LINEAR OPERATORS
BANACH CENTER PUBLICATIONS, VOLUME 38
INSTITUTE OF MATHEMATICS
POLISH ACADEMY OF SCIENCES
WARSZAWA 1997
LOG-MAJORIZATIONS AND NORM INEQUALITIES
FOR EXPONENTIAL OPERATORS
FUMIO HIAI
Department of Mathematics, Ibaraki University
Mito, Ibaraki 310, Japan
E-mail: [email protected]
Abstract. Concise but self-contained reviews are given on theories of majorization and
symmetrically normed ideals, including the proofs of the Lidskii–Wielandt and the Gelfand–
Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm
inequalities of Golden–Thompson type and its complementary type for exponential operators on
a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula
as well as for that involving operator means.
1. Introduction. Since the notion of majorization was introduced by Hardy, Littlewood, and Pólya, it has been discussed by many mathematicians in various circumstances
with various applications. First let us recall the notion of (weak) majorization in the simplest case of real vectors. For real vectors a = (a1 , . . . , an ) and b = (b1 , . . . , bn ), the weak
Pk
Pk
majorization a ≺w b means that
i=1 a[i] ≤
i=1 b[i] holds for 1 ≤ k ≤ n, where
(a[1] , . . . , a[n] ) is the decreasing rearrangement of a. The majorization a ≺ b means that
Pn
Pn
a ≺w b and i=1 ai = i=1 bi . When a and b are nonnegative, the multiplicative or
Q
logarithmic (weak) majorization can be also defined by taking product
in place of sum
P
in the above, which we referred to in [6] as the log-majorization.
Several (weak) majorizations are known for the eigenvalues and the singular values of
matrices and compact operators, as was fully clarified in Marshall and Olkin’s monograph
[62] and also in [4, 61]. These majorizations give rise to powerful devices in deriving
various norm inequalities (in particular, perturbation norm inequalities) as well as trace
or determinant inequalities for matrices or operators (see e.g. [15]). Among other things,
the Lidskii–Wielandt majorization theorem is especially famous and important. A crucial
reason why the (weak) majorization is useful in operator norm inequalities is the following
fact: For bounded Hilbert space operators A and B, the weak majorization µ(A) ≺w
1991 Mathematics Subject Classification: Primary 47A30, 47B10; Secondary 47A63, 15A42.
Research supported by Grant-in-Aid for Scientific Research 06221207.
The paper is in final form and no version of it will be published elsewhere.
[119]
120
F. HIAI
µ(B) holds if and only if kAk ≤ kBk for any symmetric (or unitarily invariant) norm
k · k, where µ(A) = (µ1 (A), µ2 (A), . . .) are the (generalized) singular values of A with
multiplicities. General theory of symmetric norms and symmetrically normed ideals was
extensively developed in Gohberg and Krein’s monumental monograph [32] (also [77,
79]). The majorization technique sometimes plays an important role in the study of
symmetrically normed ideals. For instance, the Hölder type inequality for symmetric
norms is a simple consequence of Horn’s majorization of multiplicative type.
The celebrated Golden–Thompson trace inequality, independently proved by Golden
[33], Symanzik [83], and Thompson [84], is tr eH+K ≤ tr eH eK for self-adjoint operators
(particularly for Hermitian matrices) H and K. So far, there have been many extended
or related inequalities around the Golden–Thompson inequality. For example, when H
and K are Hermitian matrices, this inequality was extended in [58, 85] to the weak
majorization µ(eH+K ) ≺w µ(eH/2 eK eH/2 ) or equivalently keH+K k ≤ keH/2 eK eH/2 k for
any unitarily invariant norm, and in [81] to tr eH+K ≤ tr(eH/n eK/n )n for all n ∈ N.
Also, the Araki–Lieb–Thirring inequality [12] (also [88]) is regarded as a strengthened
Golden–Thompson inequality and is reformulated in terms of log-majorization (see [6,
40]). On the other hand, a complementary counterpart of the Golden–Thompson trace
inequality was discovered in [44] in the course of study on lower and upper bounds for
the relative entropy, and was strengthened in [6] to the form of log-majorization by
using the technique of antisymmetric tensor powers. Restricted to the matrix case, the
above log-majorizations of Golden–Thompson type and its complementary type yield the
following norm inequalities: If H and K are Hermitian matrices and 0 < α < 1, then
k(erH/(1−α) #α erK/α )1/r k ≤ keH+K k ≤ k(erH/2 erK erH/2 )1/r k,
r > 0,
for any unitarily invariant normk·k, where #α denotes the α-power mean, i.e. the operator
mean corresponding to the operator monotone functionxα . Moreover, the above left-hand
(resp. right-hand) side increases (resp. decreases) to keH+K k as r ↓ 0.
The Golden–Thompson trace inequality was originally motivated by quantum statistical mechanics. When a Hamitonian K is given as a self-adjoint operator (assumed here to
be lower-bounded) on a Hilbert space, the partition function tr e−βK and the free energy
log tr e−βK where β is an inverse temperature constant are basically important from the
quantum statistical mechanical viewpoint. When K receives a lower-bounded perturbation by H, physicists sometimes approximate tr e−β(H+K) by tr(e−βH/n e−βK/n )n via the
Trotter product formula. Although the convergence tr(e−βH/n e−βK/n )n → tr e−β(H+K)
might have been strongly believed by physicists, there was no rigorous proof up to [40].
Indeed, it was more strongly proved in [40] that if H and K are lower-bounded self-adjoint
operators such that e−K is of trace class and H + K is essentially self-adjoint, then the
following trace norm convergence holds:
lim k(e−rH/2 e−rK e−rH/2 )1/r − e−(H+K) k1 = 0.
r↓0
(Another k · k1 -convergence under a rather strong assumption was given in [65]. Other
recent developments on the Trotter–Kato product formula in the operator norm and trace
norm are found in [37, 46, 47, 75].)
LOG-MAJORIZATIONS AND NORM INEQUALITIES
121
The present paper enjoys both aspects of a review paper and of a research paper. It
is organized in five sections which are divided into several subsections. Our main aim
is to present log-majorizations and norm inequalities for infinite-dimensional exponential
operators. For this sake, in Sections 1 and 2 we concisely review the majorization theory
and theory of symmetrically normed ideals. Although several distinguished monographs,
as cited above, are available, we intend to make the exposition completely self-contained,
so that Sections 1 and 2 may independently serve as a concise text on these subjects. The
main part of Section 1 is the proofs of the Lidskii–Wielandt and the Gelfand–Naimark
theorems for (generalized) singular values of matrices (operators). Our proofs are rather
new and based on the real interpolation method (or the K-functional method). In Section
2 we stress the majorization technique in the theory of symmetrically normed ideals.
Section 3 is taken from [40] and is not new, but we sometimes give more detailed
accounts for the convenience of the reader. We investigate log-majorizations and norm
inequalities of Golden–Thompson type for exponential operators. For instance, it is shown
that if H and K are lower-bounded self-adjoint operators, then
ke−(H +̂K) k ≤ k(e−rH/2 e−rK e−rH/2 )1/r k,
r > 0,
for any symmetric norm k · k, where H +̂K is the form sum of H and K. Preliminaries
on antisymmetric tensor powers and the Trotter–Kato exponential product formula are
included, which are quite beneficial in proving our results. Also in Subsection 3.5 we
discuss the trace norm convergence of exponential product formula together with some
technical preliminaries in the framework of von Neumann algebras.
Section 4 is considered as a complementary counterpart of Section 3. Extending the
matrix case [6, 44] (also [71]), we investigate log-majorizations and norm inequalities
involving operator means (in particular, the α-power mean), which are opposite to those
in Section 3 and considered as complementary Golden–Thompson type. If H is bounded
self-adjoint and K is lower-bounded self-adjoint, then the following is proved for any
symmetric norm k · k and 0 < α < 1:
k(e−rH/(1−α) #α e−rK/α )1/r k ≤ ke−(H+K) k,
r > 0.
The most important ingredient in the extension from the matrix case to the infinitedimensional case is an exponential product formula for operator means established in
Subsection 4.3. Finally in Section 5 we obtain further log-majorization results, for example, the log-majorization equivalent to the Furuta inequality [28], generalized logmajorizations of Horn’s type and of Golden–Thompson type, etc. Some determinant
inequalities are also included. Most results of Section 4 and many of Section 5 are new.
Although we confine ourselves to the setting of Hilbert space operators (in other words,
the setup of B(H)) in this paper, it should be mentioned that many subjects treated here
extend to the von Neumann algebra setup. In fact, based on the noncommutative integration theory ([20, 66, 78]), we can discuss the majorization theory in semifinite von
Neumann algebras (see e.g. [21, 38, 39, 41, 42, 43, 48, 49, 64]) by using the notion of
generalized s-numbers introduced in [25, 26] for measurable operators. Noncommutative
Banach function spaces (i.e. generalized symmetrically normed ideals) associated with
semifinite von Neumann algebras have been discussed in [21, 22, 53, 89, 90] for instance.
122
F. HIAI
In particular, theory of noncommutative Lp -spaces over arbitrary von Neumann algebras
was developed (e.g. the Haagerup Lp -spaces [35]). Kosaki [55] extended the Araki–Lieb–
Thirring inequality (i.e. a log-majorization result in Subsection 3.2) to the von Neumann
algebra case. Furthermore, many authors have worked on the Golden–Thompson inequality in von Neumann algebras in several ways ([8, 25, 45, 70, 76]). However, at present, the
von Neumann algebra versions for the norm convergence in Subsection 3.5 and for the
study of complementary Golden–Thompson type in Section 4 are not yet investigated.
When we want to extend the study of Section 4 to the von Neumann algebra setup, the
antisymmetric tensor technique is no longer available, so that we would have to exploit
a new method.
The contents of the paper are as follows:
1. Majorization and log-majorization
1.1. Majorization for vectors
1.2. Generalized singular values
1.3. Majorization for matrices: Lidskii–Wielandt and Gelfand–Naimark theorems
1.4. Majorization for operators
2. Symmetric norms and symmetrically normed ideals
2.1. Symmetric gauge functions and symmetric norms
2.2. Symmetrically normed ideals
2.3. Further properties of symmetric norms
2.4. Ando’s extension of Birman–Koplienko–Solomyak majorization result
3. Inequalities of Golden–Thompson type
3.1. Antisymmetric tensor powers
3.2. Araki’s log-majorization result
3.3. Trotter–Kato exponential product formula
3.4. Log-majorization and norm inequalities of Golden–Thompson type
3.5. Trace norm convergence of exponential product formula
4. Inequalities of complementary Golden–Thompson type
4.1. Preliminaries on operator means
4.2. Log-majorization for power operator means
4.3. Exponential product formula for operator means
4.4. Norm inequalities of complementary Golden–Thompson type
4.5. Norm convergence of exponential product formula for operator means
5. Miscellaneous results
5.1. Interplay between log-majorization and Furuta inequality
5.2. Other log-majorizations
5.3. Determinant inequalities
1. Majorization and log-majorization. The purpose of this section is to give a
concise but self-contained review on the majorization theory for (generalized) singular
values of matrices and operators. Complete expositions on the subject are found in [4,
62]. Also see [5] for recent developments.
1.1. Majorization for vectors. Let us start with the majorization for real vectors,
which was introduced by Hardy, Littlewood, and Pólya. For two vectors a = (a1 , . . . , an )
123
LOG-MAJORIZATIONS AND NORM INEQUALITIES
and b = (b1 , . . . , bn ) in Rn , the weak majorization a ≺w b means that
k
X
(1.1)
i=1
a[i] ≤
k
X
b[i] ,
i=1
1 ≤ k ≤ n,
where (a[1] , . . . , a[n] ) is the decreasing rearrangement of a, i.e. a[1] ≥ . . . ≥ a[n] are the
components of a in decreasing order. The majorization a ≺ b means that a ≺w b and
equality holds for k = n in (1.1). The following characterizations of majorization and
weak majorization are fundamental.
Proposition 1.1. The following conditions for a, b ∈ Rn are equivalent :
(i) a ≺ b;
Pn
Pn
(ii) i=1 |ai − r| ≤ i=1 |bi − r| for all r ∈ R;
Pn
Pn
(iii) i=1 f (ai ) ≤ i=1 f (bi ) for any convex function f on an interval containing all
ai , b i ;
(iv) a is a convex combination of coordinate permutations of b;
(v) a = Db for some doubly stochastic n × n matrix D, i.e. D = [dij ] with dij ≥ 0,
Pn
Pn
i=1 dij = 1 for 1 ≤ j ≤ n.
j=1 dij = 1 for 1 ≤ i ≤ n, and
P r o o f. (i)⇒(iv). We show that there exist a finite number of matrices D1 , . . . , DN
of the form λI + (1 − λ)Π where 0 ≤ λ ≤ 1 and Π is a permutation matrix interchanging
two coordinates only such that a = DN . . . D1 b. Then (iv) follows because DN . . . D1
becomes a convex combination of permutation matrices. We may assume that a1 ≥ . . . ≥
an and b1 ≥ . . . ≥ bn . Suppose a 6= b and choose the largest j such that aj < bj .
Then there exists k with k > j such that ak > bk . Choose the smallest such k. Let
1 − λ1 = min{bj − aj , ak − bk }/(bj − bk ) and Π1 be the permutation matrix interchanging
the jth and kth coordinates. Then 0 < λ1 < 1 because bj > aj ≥ ak > bk . Define
D1 = λ1 I + (1 − λ1 )Π1 and b(1) = D1 b. Now it is easy to check that a ≺ b(1) ≺ b and
(1)
(1)
b1 ≥ . . . ≥ bn . Moreover the jth or kth coordinates of a and b(1) are equal. When
a 6= b(1) , we can apply the above argument to a and b(1) . Repeating finite times we reach
the conclusion.
(iv)⇒(v) is trivial from the fact that any convex combination of permutation matrices
is doubly stochastic.
(v)⇒(ii). For every r ∈ R we get
n
X
i=1
|ai − r| =
n
n
n
n X
X
X
X
|bj − r|.
dij |bj − r| =
dij (bj − r) ≤
i=1
j=1
i,j=1
j=1
(ii)⇒(i). Taking large r and small r in the inequality of (ii) we have
Noting that |x| + x = 2x+ for x ∈ R where x+ = max{x, 0}, we get
(1.2)
n
n
X
X
(bi − r)+ ,
(ai − r)+ ≤
i=1
i=1
Pn
i=1
ai =
Pn
i=1 bi .
r ∈ R.
Now prove that (1.2) implies a ≺w b. When b[k] ≥ r ≥ b[k+1] ,
Pk
i=1
a[i] ≤
Pk
i−1 b[i]
follows
124
F. HIAI
because
n
X
i=1
(ai − r)+ ≥
k
X
i=1
(a[i] − r)+ ≥
k
X
i=1
a[i] − kr,
k
n
X
X
b[i] − kr.
(bi − r)+ =
i=1
i=1
PN
P
(iv)⇒(iii). Suppose that ai = k=1 λk bπk (i) , 1 ≤ i ≤ n, where λk > 0, N
k=1 λk = 1,
and πk are permutations on {1, . . . , n}. Then the convexity of f implies that
n
X
i=1
f (ai ) ≤
n X
N
X
λk f (bπk (i) ) =
i=1 k=1
n
X
f (bi ).
i=1
(iii)⇒(v) is trivial because f (x) = |x − r| is convex.
Note that (v)⇒(iv) is seen directly from the well-known theorem of Birkhoff [17] saying
that every doubly stochastic matrix is a convex combination of permutation matrices.
Proposition 1.2. The following conditions (i)–(iv) for a, b ∈ Rn are equivalent :
(i) a ≺w b;
(ii) there exists c ∈ Rn such that a ≤ c ≺ b, where a ≤ c means that ai ≤ ci ,
1 ≤ i ≤ n;
Pn
Pn
(iii) i=1 (ai − r)+ ≤ i=1 (bi − r)+ for all r ∈ R;
Pn
Pn
(iv) i=1 f (ai ) ≤
i=1 f (bi ) for any increasing convex function f on an interval
containing all ai , bi .
Moreover , if a, b ≥ 0, then the above conditions are equivalent to the following:
(v) a = Sb for some doubly substochastic n × n matrix S, i.e. S = [sij ] with sij ≥ 0,
Pn
j=1 sij ≤ 1 for 1 ≤ i ≤ n, and
i=1 sij ≤ 1 for 1 ≤ j ≤ n.
Pn
P r o o f. (i)⇒(ii). By induction on n. We may assume that a1 ≥ . . . ≥ an and b1 ≥
Pk
Pk
. . . ≥ bn . Let α = min1≤k≤n ( i=1 bi − i=1 ai ) and define ã = (a1 +α, a2 , . . . , an ). Then
Pk
Pk
a ≤ ã ≺w b and i=1 ãi = i=1 bi for some 1 ≤ k ≤ n. When k = n, a ≤ ã ≺ b. When
k < n, we get (ã1 , . . . , ãk ) ≺ (b1 , . . . , bk ) and (ãk+1 , . . . , ãn ) ≺w (bk+1 , . . . , bn ). Hence the
induction assumption implies that (ãk+1 , . . . , ãn ) ≤ (ck+1 , . . . , cn ) ≺ (bk+1 , . . . , bn ) for
some (ck+1 , . . . , cn ) ∈ Rn−k . Then a ≤ (ã1 , . . . , ãk , ck+1 , . . . , cn ) ≺ b is immediate from
ãk ≥ bk ≥ bk+1 ≥ ck+1 .
(ii)⇒(iv). Let a ≤ c ≺ b. If f is increasing and convex on an interval [α, β] containing
ai , bi , then ci ∈ [α, β] and
n
X
i=1
f (ai ) ≤
n
X
i=1
f (ci ) ≤
n
X
f (bi )
i=1
by Proposition 1.1.
(iv)⇒(iii) is trivial and (iii)⇒(i) was already shown in the proof (ii)⇒(i) of Proposition 1.1.
Now assume a, b ≥ 0 and prove (ii) ⇔ (v). If a ≤ c ≺ b, then we have, by Proposition
1.1, c = Db for some doubly stochastic matrix D and ai = αi ci for some 0 ≤ αi ≤ 1.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
125
So a = Diag(α1 , . . . , αn )Db and Diag(α1 , . . . , αn )D is a doubly substochastic matrix.
Conversely if a = Sb for a doubly substochastic matrix S, then a doubly stochastic
matrix D exists so that S ≤ D entrywise and hence a ≤ Db ≺ b.
Let a, b ∈ Rn and a, b ≥ 0. We define the weak log-majorization a ≺w(log) b when
(1.3)
k
Y
i=1
a[i] ≤
k
Y
b[i] ,
1 ≤ k ≤ n,
i=1
and the log-majorization a ≺(log) b when a ≺w(log) b and equality holds for k = n in (1.3).
It is obvious that if a and b are strictly positive, then a ≺(log) b (resp. a ≺w(log) b) if and
only if log a ≺ log b (resp. log a ≺w log b), where log a = (log a1 , . . . , log an ).
The notions of weak majorization and weak log-majorization are similarly defined for
positive bounded infinite sequences. But, to avoid discussing the decreasing rearrangement of an infinite sequence (this is considered as a special case of generalized singular
values of a bounded operator introduced later in this section), we here confine ourselves
to infinite sequences a = (a1 , a2 , . . .) and b = (b1 , b2 , . . .) such that a1 ≥ a2 ≥ . . . ≥ 0 and
P
P
b1 ≥ b2 ≥ . . . ≥ 0. For such a, b we define a ≺w b and a ≺w(log) b when ki=1 ai ≤ ki=1 bi
Qk
Qk
and i=1 ai ≤ i=1 bi , respectively, for all k ∈ N.
Proposition 1.3. Let a, b ∈ Rn with a, b ≥ 0 and suppose a ≺w(log) b. If f is a
continuous increasing function on [0, ∞) such that f (ex ) is convex , then f (a) ≺w f (b).
In particular , a ≺w(log) b implies a ≺w b. Moreover , the same assertions hold also for
infinite sequences a, b with a1 ≥ a2 ≥ . . . ≥ 0 and b1 ≥ b2 ≥ . . . ≥ 0, whenever f (0) ≥ 0
is additionally assumed.
P r o o f. First assume that a, b ∈ Rn are strictly positive and a ≺w(log) b, so that
log a ≺w log b. Since g◦h is convex when g and h are convex with g increasing, the function
(f (ex ) − r)+ is increasing and convex for any r ∈ R. Hence we get, by Proposition 1.2,
n
X
i=1
(f (ai ) − r)+ ≤
n
X
i=1
(f (bi ) − r)+ ,
which implies f (a) ≺w f (b) by Proposition 1.2 again. When a, b ≥ 0 and a ≺w(log) b, we
can choose a(m) , b(m) > 0 such that a(m) ≺w(log) b(m) , a(m) → a, and b(m) → b. Since
f (a(m) ) ≺w f (b(m) ) and f is continuous, we obtain f (a) ≺w f (b).
The case of infinite sequences is immediate from the above case. In fact, a ≺w(log) b
implies that (a1 , . . . , an ) ≺w(log) (b1 , . . . , bn ) for every n ∈ N. Hence (f (a1 ), . . . , f (an )) ≺w
(f (b1 ), . . . , f (bn )), n ∈ N, so that f (a) ≺w f (b).
1.2. Generalized singular values. In the sequel of this section, we discuss the majorization theory for singular values of matrices and operators. Our goal is to prove the
Lidskii–Wielandt and the Gelfand–Naimark theorems for generalized singular values of
bounded operators. Indeed, these theorems were proved by using the real interpolation
method in the setting of von Neumann algebras in [41, 64] (also [21]). In Subsection
1.3 we first prove the theorems for matrices by using this new method. After that, in
Subsection 1.4 we extend them from matrices to operators in a rather simple way.
126
F. HIAI
For any n × n matrix A let µ(A) = (µ1 (A), . . . , µn (A)) be the vector of singular
values of A in decreasing order, i.e. µ1 (A) ≥ . . . ≥ µn (A) are the eigenvalues of |A| =
(A∗ A)1/2 with multiplicities. When A is Hermitian, the vector of eigenvalues of A in
decreasing order is denoted by λ(A) = (λ1 (A), . . . , λn (A)). The notion of singular values
is generalized to infinite-dimensional operators. Let H be a Hilbert space (always assumed
to be separable) and B(H) the algebra of all bounded operators on H. For any A ∈ B(H)
we define the generalized singular values µ1 (A) ≥ µ2 (A) ≥ . . . of A by
T∞
µn (A) = inf{λ ≥ 0 : rank(I − E|A| (λ)) < n},
n ∈ N,
where |A| = 0 λdE|A| (λ) is the spectral decomposition of |A| so that I − E|A| (λ) is the
spectral projection of |A| corresponding to the interval (λ, ∞). The above definition of
µn (A) is a special case of the generalized s-numbers of measurable operators in the setting
of von Neumann algebras introduced in [25, 26] (also [69]). If A is a compact operator (in
particular, a matrix), then µn (A) are the usual singular values of A in decreasing order
with multiplicities.
Let µ∞ (A) = limn→∞ µn (A). Then it is easy to see that µ∞ (A) = kAke , the essential
norm of A; namely µ∞ (A) is equal to the largest α ∈ R such that E|A| (α+ε)−E|A| (α−ε)
is of infinite rank for every ε > 0. Note that µ∞ (A) = 0 if and only if A is compact. Furthermore, if µn (A) > µ∞ (A), then µn (A) is an eigenvalue of |A| with finite multiplicity.
The basic properties of µn (A) are summarized as follows. See [26] for the proof in the
von Neumann algebra setting.
Proposition 1.4. Let A, B, X, Y ∈ B(H) and n, m ∈ N.
(1) Mini-max expression:
(1.4)
µn (A) = inf{kA(I − P )k∞ : P is a projection , rank P = n − 1},
where k ·k∞ denotes the operator norm and a projection means always an orthogonal one.
Furthermore, if A ≥ 0 then
(1.5)
µn (A) = inf{
sup
hAξ, ξi : M is a subspace of H, dim M = n − 1}.
ξ∈M⊥ , kξk=1
(2) Approximation number expression:
(1.6)
µn (A) = inf{kA − Xk∞ : X ∈ B(H), rank X < n}.
(3) µ1 (A) = kAk∞ .
(4) µn (αA) = |α|µn (A) for α ∈ C.
(5) µn (A) = µn (A∗ ).
(6) If 0 ≤ A ≤ B then µn (A) ≤ µn (B).
(7) µn (XAY ) ≤ kXk∞ kY k∞ µn (A).
(8) µn+m−1 (A + B) ≤ µn (A) + µm (B).
(9) µn+m−1 (AB) ≤ µn (A)µm (B).
(10) |µn (A) − µn (B)| ≤ kA − Bk∞ .
(11) µn (f (A)) = f (µn (A)) if A ≥ 0 and f is a continuous increasing function on
[0, ∞) with f (0) ≥ 0.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
127
P r o o f. (1) Let αn be the right-hand side of (1.4). First note that this does not change
when rank P = n − 1 in (1.4) is replaced by rank P < n. So, if rank(I − E|A| (λ)) < n,
then
αn ≤ kAE|A| (λ)k∞ = k |A|E|A| (λ)k∞ ≤ λ.
Hence αn ≤ µn (A). Conversely, for any ε > 0 choose a projection P with rank P = n − 1
such that kA(I − P )k∞ < αn + ε. Suppose rank(I − E|A| (αn + ε)) ≥ n. Then there exists
ξ ∈ H with kξk = 1 such that (I − E|A| (αn + ε))ξ = ξ but P ξ = 0. This implies that
αn + ε ≤ k |A|ξk = kA(I − P )ξk < αn + ε,
a contradiction. Hence rank(I − E|A| (αn + ε)) < n and µn (A) ≤ αn + ε, implying
µn (A) ≤ αn .
When A ≥ 0, since EA1/2 (λ) = EA (λ2 ), we get µn (A) = µn (A1/2 )2 . So (1.5) follows
from (1.4), because the right-hand side of (1.5) is written as
inf{kA1/2 (I − P )k2∞ : P is a projection, rank P = n − 1}.
(2) Let βn be the right-hand side of (1.6). If rank(I − E|A| (λ)) < n, then rank(A(I −
E|A| (λ))) < n and βn ≤ kAE|A| (λ)k∞ ≤ λ. Hence βn ≤ µn (A). Conversely, if rank X < n,
then the support projection P of |X| has rank < n. Since X(I − P ) = 0, we get by (1.4)
µn (A) ≤ kA(I − P )k∞ = k(A − X)(I − P )k∞ ≤ kA − Xk∞ ,
implying µn (A) ≤ βn .
(3) is (1.4) for n = 1. (4) and (5) follow from (1.4) and (1.6), respectively. (6) is
a consequence of (1.5). It is immediate from (1.4) that µn (XA) ≤ kXk∞ µn (A). Also
µn (AY ) = µn (Y ∗ A∗ ) ≤ kY k∞ µn (A) by (5). Hence (7) holds.
Next we show (8)–(10). By (1.6), for every ε > 0, there exist X, Y ∈ B(H) with
rank X < n, rank Y < m such that kA − Xk∞ < µn (A) + ε and kB − Y k∞ < µm (B) + ε.
Since rank(X + Y ) < n + m − 1, we have
µn+m−1 (A + B) ≤ k(A + B) − (X + Y )k∞ < µn (A) + µm (B) + 2ε,
implying (8). For Z = XB + (A − X)Y we get
rank Z ≤ rank X + rank Y < n + m − 1,
kAB − Zk∞ = k(A − X)(B − Y )k∞ < (µn (A) + ε)(µm (B) + ε).
These imply (9). Letting m = 1 and replacing B by B − A in (8) we get
µn (B) ≤ µn (A) + kB − Ak∞ ,
which shows (10).
(11) Assume f (0) = 0 and let fε (t) = f (t)+εt for ε > 0. Then µn (fε (A)) = fε (µn (A))
is easily checked from Efε (A) (λ) = EA (fε−1 (λ)). Since by (10)
|µn (fε (A)) − µn (f (A))| ≤ kfε (A) − f (A)k∞ → 0
as ε ↓ 0, we have µn (f (A)) = limε↓0 fε (µn (A)) = f (µn (A)). When α = f (0) > 0, let
g(t) = f (t) − α. Then the assertion follows as
µn (f (A)) = µn (g(A) + αI) = µn (g(A)) + α = g(µn (A)) + α = f (µn (A)).
128
F. HIAI
1.3. Majorization for matrices: Lidskii–Wielandt and Gelfand–Naimark theorems.
The following majorization results are the celebrated Lidskii–Wielandt theorem for singular values of matrices as well as for eigenvalues of Hermitian matrices.
Theorem 1.5. For any n × n matrices A and B,
|µ(A) − µ(B)| ≺w µ(A − B),
that is,
k
X
j=1
|µij (A) − µij (B)| ≤
k
X
j=1
µj (A − B)
for every 1 ≤ i1 < i2 < . . . < ik ≤ n.
Theorem 1.6. For any Hermitian n × n matrices A and B,
λ(A) − λ(B) ≺ λ(A − B),
or equivalently
(λi (A) + λn−i+1 (B)) ≺ λ(A + B).
The following results due to Ky Fan are consequences of the above theorems. The
direct proofs of these weakened majorizations are much easier than those of Theorems
1.5 and 1.6.
Corollary 1.7. (1) For any matrices A and B,
µ(A + B) ≺w µ(A) + µ(B).
(2) For any Hermitian matrices A and B,
λ(A + B) ≺ λ(A) + λ(B).
The proof of Theorem 1.5 is based on the real interpolation method. We need several
lemmas. In the following, tr X denotes the usual trace of an n × n matrix X and kXk1
Pn
the trace norm of X, i.e. kXk1 = tr |X| = j=1 µj (X).
Lemma 1.8. If A and B are Hermitian, then
n
X
|λj (A) − λj (B)| ≤ kA − Bk1 .
j=1
P r o o f. Take the Jordan decomposition A − B = (A − B)+ − (A − B)− and let
C = A + (A − B)− = B + (A − B)+ . Then we have
kA − Bk1 = tr(A − B)+ + tr(A − B)−
= 2tr C − tr A − tr B
n
X
{2λj (C) − λj (A) − λj (B)}.
=
j=1
Since A ≤ C and B ≤ C, it follows from Proposition 1.4(6) that λj (A) ≤ λj (C) and
λj (B) ≤ λj (C). Hence
|λj (A) − λj (B)| ≤ 2λj (C) − λj (A) − λj (B),
which gives the conclusion.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
129
The following simple observation is due to Wielandt (see [61]).
∗
Lemma 1.9. For any n × n matrix X let X̂ = X0 X0 , a 2n × 2n Hermitian matrix.
Then
µj (X),
1 ≤ j ≤ n,
λj (X̂) =
−µ2n−j+1 (X), n + 1 ≤ j ≤ 2n.
P r o o f. We get
|X̂| =
|X|
0
,
0 |X ∗ |
1
(|X̂| + X̂) =
2
1
X̂− = (|X̂| − X̂) =
2
X̂+ =
1 |X| X ∗
,
2 X |X ∗ |
1 |X| −X ∗
.
2 −X |X ∗ |
Since λ(|X|) = λ(|X ∗ |) = µ(X), we have
λ2j−1 (|X̂|) = λ2j (|X̂|) = µj (X),
1 ≤ j ≤ n.
But since
1
1 0
X̂+
0
0 −1
0
= X̂− ,
−1
it follows that
λj (X̂+ ) = λj (X̂− ) = λ2j (|X̂|),
1 ≤ j ≤ n.
These give the assertion.
Lemma 1.10. For any A and B,
n
X
j=1
|µj (A) − µj (B)| ≤ kA − Bk1 .
∗
and B̂ = B0 B0 . Then
|A − B|
0
k − B̂k1 = tr
= 2kA − Bk1 .
0
|(A − B)∗ |
P r o o f. Let  =
A∗ A 0
0
By Lemmas 1.8 and 1.9 we have
k − B̂k1 ≥
2n
X
j=1
|λj (Â) − λj (B̂)| = 2
n
X
j=1
|µj (A) − µj (B)|.
The right-hand side of (1.7) below is known as the K-functional in the real interpolation theory.
Lemma 1.11. For any A and 1 ≤ k ≤ n,
(1.7)
k
X
j=1
µj (A) = min{kXk1 + kkY k∞ : A = X + Y }.
130
F. HIAI
P r o o f. For any decomposition A = X + Y , since µj (A) ≤ µj (X) + kY k∞ by Proposition 1.4(10), we get
k
X
j=1
µj (A) ≤
k
X
j=1
µj (X) + kkY k∞ ≤ kXk1 + kkY k∞ .
For the converse, take the polar decomposition A = V |A| and the spectral decompositon
Pn
|A| = j=1 µj (A)Pj with mutually orthogonal projections Pj of rank 1. Define
X=V
k
X
j=1
{µj (A) − µk (A)}Pj ,
n
k
o
n
X
X
µj (A)Pj .
Pj +
Y = V µk (A)
j=1
j=k+1
Then X + Y = A and
k
X
kXk1 =
j=1
µj (A) − kµk (A),
so that
kXk1 + kkY k∞ =
k
X
kY k∞ = µk (A),
µj (A).
j=1
P r o o f o f T h e o r e m 1.5. Fix k and choose, by Lemma 1.11, X and Y such that
A − B = X + Y and
k
X
µj (A − B) = kXk1 + kkY k∞ .
j=1
Let aj = µj (X + B) − µj (B) and bj = µj (A) − µj (X + B) for 1 ≤ j ≤ n. Since
aj + bj = µj (A) − µj (B), considering the diagonal matrices we have by Lemma 1.11,
k
X
j=1
|µ(A) − µ(B)|[j] =
k
X
j=1
µj (Diag(µ1 (A) − µ1 (B), . . . , µn (A) − µn (B)))
≤ kDiag(a1 , . . . , an )k1 + kkDiag(b1 , . . . , bn )k∞
n
X
|aj | + k max |bj |.
=
j=1
But Lemma 1.10 gives
n
X
j=1
|aj | =
n
X
j=1
1≤j≤n
|µj (X + B) − µj (B)| ≤ kXk1,
while by Proposition 1.4(10)
max |bj | = max |µj (A) − µj (X + B)| ≤ kA − (X + B)k∞ = kY k∞ .
1≤j≤n
1≤j≤n
Therefore
k
X
j=1
as desired.
|µ(A) − µ(B)|[j] ≤ kXk1 + kkY k∞ =
k
X
j=1
µj (A − B),
LOG-MAJORIZATIONS AND NORM INEQUALITIES
131
P r o o f o f T h e o r e m 1.6. Let A and B be Hermitian. Since
n
n
X
X
λj (A − B),
{λj (A) − λj (B)} = tr(A − B) =
j=1
j=1
it suffices to show that λ(A) − λ(B) ≺w λ(A − B). Choose a, b ∈ R such that A + aI ≥
B + bI ≥ 0. Then Theorem 1.5 gives
λ(A + aI) − λ(B + bI) ≺w λ(A − B + (a − b)I).
So the assertion follows from λ(A + aI) = λ(A) + a and analogous equalities for λ(B + bI)
and λ(A − B + (a − b)I).
Another important majorization for singular values of matrices is the Gelfand–Naimark theorem as follows.
Theorem 1.12. For any n × n matrices A and B,
(1.8)
(µi (A)µn−i+1 (B)) ≺(log) µ(AB),
or equivalently
(1.9)
k
Y
j=1
µij (AB) ≤
k
Y
j=1
{µj (A)µij (B)}
for all 1 ≤ i1 < i2 < . . . < ik ≤ n.
An immediate corollary of this theorem is the majorization result due to Horn.
Corollary 1.13. For any matrices A and B,
µ(AB) ≺(log) µ(A)µ(B),
where µ(A)µ(B) = (µi (A)µi (B)).
The following proof of Theorem 1.12 is a modification of [64] where (1.9) was proved
in the von Neumann algebra setting.
Lemma 1.14. Assume that A is invertible. Then for any 1 ≤ k ≤ n, there exist X and
Y such that X ≥ I, A = Y X, and
k
Y
j=1
µj (A) = det X · kY kk∞ ,
where det X denotes the determinant of X.
Pn
P r o o f. Let A = V |A| be the polar decomposition and |A| =
j=1 µj (A)Pj the
spectral decomposition. Define
k
n
n
k
o
n
X
X
X
X
µj (A)
X=
Pj +
µj (A)Pj .
Pj +
Pj ,
Y = V µk (A)
µ (A)
j=1 k
j=1
j=k+1
j=k+1
Then we have X ≥ I and Y X = A. Moreover,
det X · kY kk∞ =
k
k
Y
Y
µj (A)
µj (A).
· µk (A)k =
µ (A)
j=1 k
j=1
132
F. HIAI
P r o o f o f T h e o r e m 1.12. To prove (1.8) and (1.9), we may assume by Proposition
1.14(10) that A and B are invertible. Note that
n
Y
{µi (A)µn−i+1 (B)} = | det A · det B| =
i=1
n
Y
µi (AB).
i=1
Hence (1.8) is equivalent to (1.9) if we replace A and B by AB and B −1 , respectively, in
(1.8). Now let 1 ≤ i1 < . . . < ik ≤ n and X, Y be as in Lemma 1.14. Since
µi (AB) = µi (Y XB) ≤ kY k∞ µi (XB),
1 ≤ i ≤ n,
we get
k
Y
j=1
µij (AB) ≤ kY kk∞ ·
k
Y
µij (XB).
j=1
On the other hand, since by Proposition 1.4,
µi (XB) = µi (B ∗ X 2 B)1/2 ≥ µi (B ∗ B)1/2 = µi (B),
1 ≤ i ≤ n,
we get
k
n
Y
µij (XB) Y µi (XB)
det |XB|
≤
=
= det X.
(B)
µ
µ
(B)
det |B|
i
i
j
j=1
i=1
Therefore
k
Y
j=1
µij (AB) ≤ det X · kY kk∞ ·
k
Y
µij (B) =
k
Y
{µj (A)µij (B)},
j=1
j=1
showing (1.9).
R e m a r k 1.15. The above proof is a multiplicative or logarithmic counterpart of the
real interpolation method in the proof of Theorem 1.5. As shown in [64], for 1 ≤ k ≤ n
we have
k
Y
µj (A) = min{k log |X| k1 + k log kY k∞ : A = Y X}
(1.10)
log
j=1
whenever µk (A) > 0. In fact, if A = Y X then
log
k
Y
j=1
µj (A) ≤
k
X
j=1
log{kY k∞ µj (X)} ≤
k
X
j=1
λj (log |X|) + k log kY k∞
≤ k log |X| k1 + k log kY k∞ .
This together with the proof of Lemma 1.14 shows (1.10).
1.4. Majorization for operators. In this subsection let us extend Theorems 1.5 and
1.12 to generalized singular values µ(A) = (µ1 (A), µ2 (A), . . .) of infinite-dimensional operators. The next lemma is useful for this sake.
Lemma 1.16. Let A ∈ B(H), n ∈ N, and ε > 0. Then there exists a projection P of
finite rank such that
µi (P AP ) ≥ (µi (A) − ε)+ ,
1 ≤ i ≤ n.
133
LOG-MAJORIZATIONS AND NORM INEQUALITIES
Furthermore, there exists an orthonormal set {ξ1 , . . . , ξn } such that
k |A|ξi − µi (A)ξi k ≤ ε,
1 ≤ i ≤ n.
P r o o f. When µm (A) > 0 = µm+1 (A) for some 1 ≤ m < n, we may consider
(µ1 (A), . . . , µm (A)) instead of (µ1 (A), . . . , µn (A)). So it suffices to assume that µn (A) >
ε > 0. Let α = µ∞ (A) (= limn→∞ µn (A)). Suppose that µn (A) = α and µk−1 (A) >
α = µk (A) = . . . = µn (A) for some 1 ≤ k ≤ n. Then all µi (A), 1 ≤ i < k (if any),
are eigenvalues of |A| and E|A| (α) − E|A| (α − ε) is of infinite rank. So we can choose an
orthonormal set {ξ1 , . . . , ξn } such that |A|ξi = µi (A)ξi for 1 ≤ i < k and ξi , k ≤ i ≤ n, are
in the range of E|A| (α) − E|A| (α − ε). Let P0 and P be the projections onto the subspaces
spanned by {ξ1 , . . . , ξn } and {ξ1 , . . . , ξn , Aξ1 , . . . , Aξn }, respectively. For 1 ≤ i ≤ n we
get
µi (P AP ) ≥ µi ((P AP )P0 ) = µi (AP0 ) = µi (P0 |A|2 P0 )1/2 ≥ α − ε.
For 1 ≤ i < k, since µi (P0 |A|2 P0 )1/2 = µi (A), we get µi (P AP ) = µi (A). Hence the
required condition is fulfilled. When µn (A) > α, all µi (A), 1 ≤ i ≤ n, are eigenvalues
of |A| and the proof is done as above. Also the second assertion follows from the above
proof.
Theorem 1.17. If A, B ∈ B(H), then
k
X
(1.11)
j=1
|µij (A) − µij (B)| ≤
k
X
j=1
µj (A − B)
for all 1 ≤ i1 < i2 < . . . < ik . In particular ,
(1.12)
µ(A + B) ≺w µ(A) + µ(B),
A, B ∈ B(H).
P r o o f. Let 1 ≤ i1 < . . . < ik = n. For any ε > 0 there exists, by Lemma 1.16,
projections P, Q of finite rank such that µi (P AP ) ≥ (µi (A) − ε)+ and µi (QBQ) ≥
(µi (B) − ε)+ for 1 ≤ i ≤ n. Let E = P ∨ Q, a projection of finite rank. Then for
1 ≤ i ≤ n,
µi (A) ≥ µi (EAE) ≥ µi (P AP ) ≥ (µi (A) − ε)+
and µi (B) ≥ µi (EBE) ≥ (µi (B) − ε)+ . Applying Theorem 1.5 to EAE, EBE ∈ B(EH)
(considered as matrices), we have
k
X
j=1
|µij (EAE) − µij (EBE)| ≤
k
X
j=1
µj (E(A − B)E) ≤
k
X
j=1
µj (A − B).
Letting ε ↓ 0 we obtain (1.11), which implies (1.12) by letting ij = j and replacing A by
A + B.
Theorem 1.18. If A, B ∈ B(H), then
(1.13)
k
Y
j=1
µij (AB) ≤
k
Y
j=1
{µj (A)µij (B)}
for all 1 ≤ i1 < i2 < . . . < ik . In particular ,
(1.14)
µ(AB) ≺w(log) µ(A)µ(B),
A, B ∈ B(H).
134
F. HIAI
P r o o f. Let 1 ≤ i1 < . . . < ik = n and ε > 0. By Lemma 1.16 there exists a
projection P of finite rank such that µi (P ABP ) ≥ (µi (AB) − ε)+ for 1 ≤ i ≤ n. Let
E be the projection onto the subspace spanned by P H ∪ BP H, which is of finite rank.
Then P ABP = (P AE)(EBP ). Applying (1.9) to P AE, EBP ∈ B(EH), we have
k
X
j=1
µij (P ABP ) ≤
k
Y
j=1
{µj (P AE)µij (EBP )} ≤
Y
{µj (A)µij (B)},
which shows (1.13) as ε ↓ 0.
2. Symmetric norms and symmetrically normed ideals. This section is a selfcontained review on symmetric norms and symmetrically normed ideals. Our exposition
is somewhat restricted to the material which will be necessary in the subsequent sections.
See [32, 79] (also [77]) for full theory on the subject.
2.1. Symmetric gauge functions and symmetric norms. Let sfin denote the linear space
of all infinite sequences of real numbers having only finitely many nonzero terms. A norm
Φ on sfin is called to be symmetric if Φ satisfies
(2.1)
Φ(a1 , a2 , . . .) = Φ(ε1 aπ(1) , ε2 aπ(2) , . . .)
for any permutation π on N and εi = ±1. This condition is equivalently written as
Φ(a1 , a2 , . . .) = Φ(a∗1 , a∗2 , . . .)
where (a∗1 , a∗2 , . . .) is the decreasing rearrangement of (|a1 |, |a2 |, . . .). A symmetric norm
on sfin is called a symmetric gauge function.
Lemma 2.1. Let Φ be a symmetric gauge function.
(1) If (ai ), (bi ) ∈ sfin and |ai | ≤ |bi | for i ∈ N, then
Φ(a1 , a2 , . . .) ≤ Φ(b1 , b2 , . . .).
(2) Under the normalization Φ(1, 0, 0, . . .) = 1,
X
|ai |,
sup |ai | ≤ Φ(a1 , a2 , . . .) ≤
i
i
(ai ) ∈ sfin ,
that is, the ℓ∞ -norm (resp. ℓ1 -norm) is the least (resp. greatest ) symmetric gauge function.
P r o o f. (1) In view of (2.1) we may show that for 0 ≤ α ≤ 1,
Φ(αa1 , a2 , a3 , . . .) ≤ Φ(a1 , a2 , a3 , . . .).
This is seen as follows:
Φ(αa1 , a2 , a3 , . . .)
1+α
1−α
1+α
1−α
1+α
1−α
=Φ
a1 +
(−a1 ),
a2 +
a2 ,
a3 +
a3 , . . .
2
2
2
2
2
2
1−α
1+α
Φ(a1 , a2 , a3 , . . .) +
Φ(−a1 , a2 , a3 , . . .) = Φ(a1 , a2 , a3 , . . .).
≤
2
2
(2) Since by (2.1) and (1),
|ai | = Φ(ai , 0, 0, . . .) ≤ Φ(a1 , . . . , ai , . . .),
LOG-MAJORIZATIONS AND NORM INEQUALITIES
135
the first inequality holds. The second follows from
Φ(a1 , . . . , an , 0, 0, . . .) ≤
n
X
Φ(ai , 0, 0, . . .) =
n
X
i=1
i=1
|ai |.
Lemma 2.2. If (a1 , . . . , an ), (b1 , . . . , bn ) ∈ Rn and (|a1 |, . . . , |an |) ≺w (|b1 |, . . . , |bn |),
then
Φ(a1 , . . . , an , 0, 0, . . .) ≤ Φ(b1 , . . . , bn , 0, 0, . . .).
P r o o f. By Proposition 1.2 there exists (c1 , . . . , cn ) ∈ Rn such that
(|a1 |, . . . , |an |) ≤ (c1 , . . . , cn ) ≺ (|b1 |, . . . , |bn |).
Proposition 1.1 says that (c1 , . . . , cn ) is a convex combination of coordinate permutations
of (|b1 |, . . . , |bn |). This implies by Lemma 2.1(1) and (2.1) that
Φ(a1 , . . . , an , 0, 0, . . .) ≤ Φ(c1 , . . . , cn , 0, 0, . . .) ≤ Φ(b1 , . . . , bn , . . .).
Let C(H) be the algebra of all compact operators on a separable Hilbert space H. We
denote by Cfin (H) the set of all operators of finite rank on H. A norm k · k on Cfin (H) is
called to be unitarily invariant if
kU AV k = kAk
for every A ∈ Cfin (H) and any unitaries U, V on H. A unitarily invariant norm on Cfin (H) is
also called a symmetric norm. The following fundamental theorem is due to von Neumann
[67] (also [63, 77]).
Theorem 2.3. There is a bijective correspondence between symmetric gauge functions
Φ and unitarily invariant norms k · k on Cfin (H) which is determined by the formula
(2.2)
kAk = Φ(µ1 (A), µ2 (A), . . .),
A ∈ Cfin (H).
Furthermore, a norm k · k on Cfin (H) is unitarily invariant if and only if
(2.3)
kXAY k ≤ kXk∞ kY k∞ kAk
for every A ∈ Cfin (H) and X, Y ∈ B(H).
P r o o f. Suppose that Φ is a symmetric gauge function. Define k · k on Cfin (H) by the
formula (2.2). Let A, B ∈ Cfin (H). Since µ(A + B), µ(A) + µ(B) ∈ sfin and µ(A + B) ≺w
µ(A) + µ(B) by (1.12) (or Corollary 1.7(1)), we get, by Lemma 2.2,
kA + Bk ≤ Φ(µ1 (A) + µ1 (B), µ2 (A) + µ2 (B), . . .)
≤ Φ(µ1 (A), µ2 (A), . . .) + Φ(µ1 (B), µ2 (B), . . .) = kAk + kBk.
Also it is clear that kAk = 0 if and only if µ1 (A) = 0 or A = 0. For α ∈ C we get by
Proposition 1.4(4)
kαAk = Φ(|α|µ1 (A), |α|µ2 (A), . . .) = |α| kAk.
Hence k · k is a norm on Cfin (H), which is unitarily invariant because µn (U AV ) = µn (A)
for all unitaries U, V .
136
F. HIAI
Conversely, suppose that k · k is a unitarily invariant norm on Cfin (H). Choose an
orthonormal basis {ϕi } of H and define Φ : sfin → R by
X
(ai ) ∈ sfin ,
ai ϕi ⊗ ϕi ,
Φ(a1 , a2 , . . .) = i
where ϕ ⊗ ψ denotes the Schatten form, i.e. (ϕ ⊗ ψ)ξ = hξ, ψiϕ for ϕ, ψ, ξ ∈ H. Then it
is immediate that Φ is a norm on sfin . For any permutation π on N and εi = ±1, we can
define unitaries U, V on H by U ϕπ(i) = εi ϕi and V ϕπ(i) = ϕi for i ∈ N, so that
X
X
aπ(i) U ϕπ(i) ⊗ V ϕπ(i) aπ(i) ϕπ(i) ⊗ ϕπ(i) V ∗ = Φ(a1 , a2 , . . .) = U
i
X i
εi aπ(i) ϕi ⊗ ϕi = Φ(ε1 aπ(i) , ε2 aπ(2) , . . .).
=
i
P
Hence Φ is a symmetric gauge function. Any A ∈ Cfin (H) is written as A = ni=1 µi (A)ξi ⊗
ηi for some orthonormal sets {ξ1 , . . . , ξn } and {η1 , . . . , ηn }. Take unitaries U, V such that
U ϕi = ξi , V ϕi = ηi for 1 ≤ i ≤ n. Then we get
n
n
X
X
µi (A)ϕi ⊗ ϕi V ∗ = kAk,
µi (A)ϕi ⊗ ϕi = U
Φ(µ1 (A), µ2 (A), . . .) = i=1
i=1
so that (2.2) holds. Therefore the first assertion is proved.
It is clear that a norm k · k on Cfin (H) is unitarily invariant if it satisfies (2.3). Conversely, if k · k is a unitarily invariant norm with the corresponding gauge function Φ,
then by Proposition 1.4(7),
kXAY k ≤ Φ(kXk∞ kY k∞ µ1 (A), kXk∞ kY k∞ µ2 (A), . . .) = kXk∞ kY k∞ kAk
for A ∈ Cfin (H) and X, Y ∈ B(H).
2.2. Symmetrically normed ideals. Let Φ be a symmetric gauge function. When a =
(a1 , a2 , . . .) is a bounded real sequence, we define
Φ(a) = sup Φ(a1 , . . . , an , 0, 0, . . .) ∈ [0, ∞].
n
Let sΦ be the set of all bounded real sequences a with Φ(a) < ∞. Moreover, extending
k · k on Cfin (H), we define for any A ∈ B(H),
(2.4)
kAk = sup Φ(µ1 (A), . . . , µn (A), 0, 0, . . .) ∈ [0, ∞].
n
Let CΦ (H) denote the set of all A ∈ B(H) with kAk < ∞, i.e.
µ(A) = (µ1 (A), µ2 (A), . . .) ∈ sΦ .
In this way, a symmetric norm k · k on Cfin (H) can extend to all operators in B(H)
permitting ∞. Then we have:
Proposition 2.4. Let A, B, X, Y ∈ B(H) and k · k be a symmetric norm.
(1)
(2)
(3)
(4)
kAk = kA∗ k.
kXAY k ≤ kXk∞ kY k∞ kAk.
If µ(A) ≺w µ(B) (in particular , if |A| ≤ |B|), then kAk ≤ kBk.
If µ(A) ≺w µ(B) and B ∈ C(H), then A ∈ C(H).
137
LOG-MAJORIZATIONS AND NORM INEQUALITIES
(5) Under the normalization Φ(1, 0, 0, . . .) = 1 (or kP k = 1 for a projection of rank
one), kAk∞ ≤ kAk ≤ kAk1 .
P r o o f. (1) and (2) immediately follow from definition (2.4), Lemma 2.1(1), and the
corresponding properties of Proposition 1.4. If µ(A) ≺w µ(B) then Lemma 2.2 gives
Φ(µ1 (A), . . . , µn (A), 0, 0, . . .) ≤ Φ(µ1 (B), . . . , µn (B), 0, 0, . . .),
n ∈ N,
and so kAk ≤ kBk. Hence (3) holds. (4) is seen from the fact that A ∈ C(H) if and
Pn
only if µn (A) → 0, which is equivalent to n−1 i=1 µi (A) → 0. (5) follows from Lemma
2.1(2).
Theorem 2.5. Let Φ be a symmetric gauge function.
(1) CΦ (H) is a Banach space in the norm (2.4) and is a two-sided ideal of B(H).
(2) If Φ is inequivalent to the ℓ∞ -norm, then CΦ (H) ⊂ C(H).
P r o o f. (1) Let A, B ∈ CΦ (H). For every n ∈ N, since (1.12) gives
(µ1 (A + B), . . . , µn (A + B)) ≺w (µ1 (A) + µ1 (B), . . . , µn (A) + µn (B)),
we have by Lemma 2.2,
Φ(µ1 (A + B), . . . , µn (A + B), 0, 0, . . .)
≤ Φ(µ1 (A), . . . , µn (A), 0, 0, . . .) + Φ(µ1 (B), . . . , µn (B), 0, 0, . . .) ≤ kAk + kBk.
This implies that A + B ∈ CΦ (H) and kA + Bk ≤ kAk + kBk. Now it is easily verified
that CΦ (H) is a linear space and k · k is a norm on CΦ (H). To prove the completeness, let
{Ak } be a Cauchy sequence in CΦ (H). Since {Ak } is k · k∞ -Cauchy by Proposition 2.4(5),
there exists A ∈ B(H) such that kAk − Ak∞ → 0. For every n ∈ N, since
Φ(µ1 (Ak − Am ), . . . , µn (Ak − Am ), 0, 0, . . .) ≤ kAk − Am k
and µi (Ak − Am ) → µi (Ak − A) as m → ∞ thanks to Proposition 1.4(10), we get
Φ(µ1 (Ak − A), . . . , µn (Ak − A), 0, 0, . . .) ≤ lim kAk − Am k.
m→∞
Therefore kAk − Ak ≤ limm kAk − Am k, which implies that A ∈ CΦ (H) and kAk − Ak → 0
as k → ∞. It is immediate from Proposition 2.4(2) that CΦ (H) is a two-sided ideal of
B(H).
(2) Suppose that CΦ (H) contains a non-compact operator A. Then we have α =
inf n µn (A) > 0 and
αΦ(1, 1, 1, . . .) = Φ(α, α, α, . . .) ≤ Φ(µ1 (A), µ2 (A), . . .) = kAk < ∞,
so that Φ(a1 , a2 , . . .) ≤ α−1 kAk supn |an |. This together with Lemma 2.1(2) implies that
Φ is equivalent to the ℓ∞ -norm.
(0)
The Banach space CΦ (H) as well as CΦ (H) defined below is called a symmetrically
normed ideal associated with a symmetric gauge function Φ. It is seen as in Theorem
2.5 that sΦ is a Banach space in the norm Φ. In fact, if {ϕi } is an orthonormal basis
P
of H, then it is not difficult to show that Φ(a) = k i ai ϕi ⊗ ϕi k for any bounded real
P
sequence a = (ai ) and sΦ is isometrically imbedded in CΦ (H) by a ∈ sΦ 7→ i ai ϕi ⊗ ϕi
(0)
whose range is a closed (real) subspace of CΦ (H). Let sΦ be the closure of sfin in sΦ .
138
F. HIAI
(0)
(0)
(0)
We denote by CΦ (H) the set of all A ∈ B(H) such that µ(A) ∈ sΦ . When sΦ = sΦ or
(0)
CΦ (H) = CΦ (H), Φ is called regular.
(0)
Theorem 2.6. Let Φ, CΦ (H), and CΦ (H) be as above.
(0)
(0)
(1) CΦ (H) is the closure of Cfin (H) in CΦ (H) and CΦ (H) ⊂ C(H).
(0)
(2) CΦ (H) is a Banach space in the norm k · k of CΦ (H) and is a two-sided ideal of
B(H).
(0)
P r o o f. (1) It is clear that Cfin (H) ⊂ CΦ (H). It follows from Lemma 2.1(2) that if
(0)
a = (a1 , a2 , . . .) ∈ sΦ , then a belongs to the ℓ∞ -norm closure of sfin so that an → 0.
P
(0)
(0)
This shows that CΦ (H) ⊂ C(H). So each A ∈ CΦ (H) is written as A = i µi (A)ξi ⊗ ηi
with orthonormal sequences {ξi } and {ηi }. Then
∞
n
X
X
µi (A)ξi ⊗ ηi = Φ(µn+1 (A), µn+2 (A), . . .).
µi (A)ξi ⊗ ηi = A −
i=n+1
i=1
(0)
But it is easily seen from µ(A) ∈ sΦ that
lim Φ(µn+1 (A), µn+2 (A), . . .) = 0.
n→∞
Hence A is in the closure of Cfin (H). Conversely, let A be in the closure of Cfin (H). For
any ε > 0 there exists B ∈ Cfin (H) such that kA − Bk < ε. Since (1.11) says that for
every n ∈ N,
(|µ1 (A) − µ1 (B)|, . . . , |µn (A) − µn (B)|) ≺w (µ1 (A − B), . . . , µn (A − B)),
it follows from Lemma 2.2 that
Φ(µ1 (A) − µ1 (B), µ2 (A) − µ2 (B), . . .) ≤ kA − Bk < ε.
(0)
(0)
Since µ(B) ∈ sfin , we have µ(A) ∈ sΦ and so A ∈ CΦ (H).
(2) The first assertion follows from (1). The second is easily verified by using Proposition 2.4(2).
For instance, let Φp be the ℓp -norm and k · kp the corresponding symmetric norm
where 1 ≤ p ≤ ∞. When 1 ≤ p < ∞ we have CΦp (H) = Cp (H), the Schatten p-class.
More generally, for 0 < p < ∞ we can define the p-class Cp (H) as the space of all A ∈ C(H)
such that
o1/p
nX
µi (A)p
< ∞.
kAkp = (tr |A|p )1/p =
i
But when 0 < p < 1, k · kp is not a norm but a quasi-norm. In particular, C1 (H) is the
trace class and C2 (H) is the Hilbert-Schmidt class. When p = ∞ we have CΦ∞ (H) = B(H)
(0)
and CΦ∞ (H) = C(H). Note that kAkp ≥ kAkq and Cp (H) ⊂ Cq (H) if 0 < p < q ≤ ∞.
Another important class of symmetric norms is the Ky Fan norms k · k(k) defined by
kAk(k) =
k
X
i=1
µi (A),
k ∈ N.
Obviously, all k · k(k) are equivalent to k · k∞ and k · k(1) = k · k∞ . See [32] for more delicate
examples of symmetrically normed ideals such as Macaev ideals.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
139
Here we show the Hölder inequality to illustrate the usefulness of the majorization
technique.
Proposition 2.7. Let 0 < p, p1 , p2 ≤ ∞ and 1/p = 1/p1 + 1/p2 . Then
kABkp ≤ kAkp1 kBkp2 ,
A, B ∈ B(H).
Hence AB ∈ Cp (H) if A ∈ Cp1 (H) and B ∈ Cp2 (H).
P r o o f. Suppose that 0 < p1 , p2 < ∞, because the result is obvious by Proposition
1.4(7) when p1 = ∞ or p2 = ∞. Since by (1.14)
(µi (AB)p ) ≺w(log) (µi (A)p µi (B)p ),
it follows from Proposition 1.3 that
(µi (AB)p ) ≺w (µi (A)p µi (B)p ).
Since (p1 /p)−1 + (p2 /p)−1 = 1, the usual Hölder inequality for vectors shows that for
every n ∈ N,
n
n
o1/p
o1/p nX
nX
µi (A)p µi (B)p
≤
µi (AB)p
i=1
i=1
≤
n
nX
µi (A)p1
i=1
n
o1/p1 nX
µi (B)p2
i=1
o1/p2
≤ kAkp1 kBkp2 .
This yields the conclusion.
Corresponding to each symmetric gauge function Φ, let us define Φ′ : sfin → R by
o
nX
ai bi : a ∈ sfin , Φ(a) ≤ 1 .
Φ′ (b1 , b2 , . . .) = sup
i
Then it is an easy task to check that Φ′ is again a symmmetric norm on sfin . The symmetric gauge function Φ′ is said to be conjugate to Φ. Note that Φ′′ = Φ. For example,
when 1 ≤ p ≤ ∞ and 1/p + 1/q = 1, the ℓp -norm is conjugate to the ℓq -norm.
The following generalized Hölder inequality can be shown as Proposition 2.7.
Lemma 2.8. Let Φ, Φ1 , Φ2 be symmetric gauge functions with the corresponding norms
k · k, ||| · |||1 , ||| · |||2 . If
Φ(a1 b1 , a2 b2 , . . .) ≤ Φ1 (a)Φ2 (b),
a, b ∈ sfin ,
then
′
kABk ≤ |||A|||1 |||B|||2 ,
A, B ∈ B(H).
In particular , if k · k is the symmetric norm corresponding to Φ′ conjugate to Φ, then
kABk1 ≤ kAk kBk′ for every A, B ∈ B(H). Hence AB ∈ C1 (H) if A ∈ CΦ (H) and
B ∈ CΦ′ (H).
P r o o f. By (1.14), Proposition 1.3, and Lemma 2.2, we have for every n ∈ N
Φ(µ1 (AB), . . . , µn (AB), 0, 0, . . .)
≤ Φ(µ1 (A)µ1 (B), . . . , µn (A)µn (B), 0, 0, . . .)
≤ Φ1 (µ1 (A), . . . , µn (A), 0, 0, . . .)Φ2 (µ1 (B), . . . , µn (B), 0, 0, . . .) ≤ |||A|||1 |||B|||1 ,
140
F. HIAI
showing the first assertion. For the second part, note by definition of Φ′ that
X
|ai bi | ≤ Φ(a)Φ′ (b), a, b ∈ sfin .
i
Theorem 2.9. If Φ and Φ′ are conjugate symmetric gauge functions, then the dual
(0)
(0)
Banach space CΦ (H)∗ of CΦ (H) is isometrically isomorphic to CΦ′ (H) by the duality
(0)
(A, B) 7→ tr(AB) for A ∈ CΦ (H) and B ∈ CΦ′ (H), where tr denotes the usual trace on
C1 (H).
P r o o f. For any B ∈ CΦ′ (H) we can define, by Lemma 2.8, a linear functional fB :
→ C by fB (A) = tr(AB). Since
(0)
CΦ
|tr(AB)| ≤ kABk1 ≤ kAk kBk′ ,
(0)
(0)
A ∈ CΦ (H),
(0)
it follows that fB ∈ CΦ (H)∗ and kfB k ≤ kBk′ . Now let f ∈ CΦ (H)∗ and consider a
sesqui-linear form (ξ, η) 7→ f (ξ ⊗ η), ξ, η ∈ H. Since
|f (ξ ⊗ η)| ≤ kf k kξ ⊗ ηk = kf kΦ(1, 0, 0, . . .)kξk kηk,
there exists B ∈ B(H) such that hBξ, ηi = f (ξ ⊗ η) for all ξ, η ∈ H. For each n ∈ N and
ε > 0, by Lemma 1.16 there exists a projection P of finite rank such that µi (P BP ) ≥
(µ (B) − ε)+ for 1 ≤ i ≤ n and also m = rank P ≥ n. We write P BP as P BP =
Pim
i=1 µi (P BP )ξi ⊗ ηi with orthonormal bases {ξi } and {ηi } of P H. Since
µi (P BP ) = hP BP ηi , ξi i = hBηi , ξi i = f (ηi ⊗ ξi ),
if a ∈ sfin and Φ(a) ≤ 1, then
n
X
ai µi (P BP ) = f
i=1
n
X
i=1
n
X
ai ηi ⊗ ξi ai ηi ⊗ ξi ≤ kf k i=1
= kf kΦ(a1 , . . . , an , 0, 0, . . .) ≤ kf k.
This shows that
Φ′ (µ1 (P BP ), . . . , µn (P BP ), 0, 0, . . .) ≤ kf k.
Letting ε ↓ 0 we get
Φ′ (µ1 (B), . . . , µn (B), 0, 0, . . .) ≤ kf k,
n ∈ N,
so that B ∈ CΦ′ (H) and kBk′ ≤ kf k. Since f (A) = fB (A) for A ∈ Cfin (H) and Cfin (H)
(0)
is dense in CΦ (H), we have f = fB and hence kfB k = kBk′ . Thus B ∈ CΦ′ (H) 7→ fB ∈
(0)
CΦ (H)∗ is a surjective isometry.
∼ B(H) and Cp (H)∗ ∼
As special cases we have C1 (H)∗ =
= Cq (H) when 1 < p < ∞,
1/p + 1/q = 1. The above theorem shows that CΦ (H)∗ ∼
C
= Φ′ (H) if Φ is regular and that
′
CΦ (H) is reflexive if and only if both Φ and Φ are regular.
2.3. Further properties of symmetric norms. In this subsection let us present, for later
use, some further results concerning symmetric norms. The close relation between the
(log-)majorization and the symmetric norm inequalities is summarized in the following
proposition.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
141
Proposition 2.10. Consider the following conditions for A, B ∈ B(H). Then:
(i)⇔(ii)⇒(iii)⇔(iv)⇔(v)⇔(vi).
(i) µ(A) ≺w(log) µ(B);
(ii) kf (|A|)k ≤ kf (|B|)k for every symmetric norm k · k and every continuous increasing function f on [0, ∞) such that f (0) ≥ 0 and f (ex ) is convex ;
(iii) µ(A) ≺w µ(B);
(iv) kAk(k) ≤ kBk(k) for every k ∈ N;
(v) kAk ≤ kBk for every symmetric norm k · k;
(vi) kf (|A|)k ≤ kf (|B|)k for every symmetric norm k · k and every increasing convex
function f on [0, ∞) such that f (0) ≥ 0.
P r o o f. (i)⇒(ii). Let f be as in (ii). By Propositions 1.3 and 1.4(11) we have
(2.5)
µ(f (|A|)) = f (µ(A)) ≺w f (µ(B)) = µ(f (|B|)).
This implies by Proposition 2.4(3) that kf (|A|)k ≤ kf (|B|)k for any symmetric norm.
(ii)⇒(i). Take k · k = k · k(k) , the Ky Fan norms, and f (x) = log(1 + ε−1 x) for ε > 0.
Then f satisfies the condition in (ii). Since
µi (f (|A|)) = f (µi (A)) = log(ε + µi (A)) − log ε,
kf (|A|)k(k) ≤ kf (|B|)k(k) means that
k
k
Y
Y
(ε + µi (B)).
(ε + µi (A)) ≤
i=1
i=1
Qk
Qk
Letting ε ↓ 0 we get i=1 µi (A) ≤ i=1 µi (B) and hence (i) follows.
(iii) ⇔ (iv) is trivial by definition of k·k(k) and (vi)⇒(v)⇒(iv) is clear. Finally assume
(iii) and let f be as in (vi). Proposition 1.2 yields (2.5) again, so that (vi) follows. Hence
(iii)⇒(vi) holds.
The following is a noncommutative analogue of Fatou’s lemma.
Proposition 2.11. Any symmetric norm k · k given by (2.4 ) is lower-semicontinuous
in WOT (i.e. the weak operator topology) on B(H).
P r o o f. Let Φ be the symmetric gauge function for k · k. Using (2.4), Lemma 1.16,
and Theorem 2.9, we have for every A ∈ B(H),
kAk = sup{kP AP k : P is a projection of finite rank}
(0)
= sup{|tr(XP AP )| : X ∈ CΦ′ (H), kXk′ ≤ 1, P is a projection of finite rank}.
Hence the assertion follows because A 7→ tr(XP AP ) is continuous in WOT whenever P
is of finite rank.
Let B(H)+ denote the set of positive operators in B(H). The next proposition extending [79, Theorem 2.16] is a noncommutative variant of the dominated convergence
theorem.
Proposition 2.12. Let Aj , A ∈ B(H) and B ∈ B(H)+ . Let Φ be a symmetric gauge
function with the corresponding norm k · k. Assume that |Aj | ≤ B and |A∗j | ≤ B for
142
F. HIAI
(0)
all j as well as |A| ≤ B and |A∗ | ≤ B. If Aj → A in WOT and B ∈ CΦ (H), then
kAj − Ak → 0.
For the proof we need:
Lemma 2.13. Let Φ and k · k be as above.
(1) If 1 < p < ∞ and 1/p + 1/q = 1, then
Φ(a1 b1 , a2 b2 , . . .) ≤ Φ(|a1 |p , |a2 |p , . . .)1/p Φ(|b1 |q , |b2 |q , . . .)1/q ,
a, b ∈ sfin .
(2) For every 1 < p < ∞ define
(p)
Φ(p) (a1 , a2 , . . .) = Φ(|a1 |p , |a2 |p , . . .)1/p ,
a ∈ sfin .
Then Φ
is a symmetric gauge function and the corresponding symmetric norm is
k | · |p k1/p .
P r o o f. (1) We may assume that a, b ≥ 0. From the Young inequality ai bi ≤ p−1 api +
Lemma 2.1(1) gives
1
1
Φ(a1 b1 , a2 b2 , . . .) ≤ Φ(ap1 , ap2 , . . .) + Φ(bq1 , bq2 , . . .).
p
q
q −1 bqi ,
Replacing a, b by αa, α−1 b with α > 0, we get
α−q
αp
Φ(ap1 , ap2 , . . .) +
Φ(bq1 , bq2 , . . .).
Φ(a1 b1 , a2 b2 , . . .) ≤
p
q
Then the desired inequality follows by minimizing the above right-hand side in α > 0.
(2) Let 1/p + 1/q = 1 and a, b ∈ sfin . We have
Φ(|a1 + b1 |p , |a2 + b2 |p , . . .) ≤ Φ(|a1 | |a1 + b1 |p−1 , |a2 | |a2 + b2 |p−1 , . . .)
+ Φ(|b1 | |a1 + b1 |p−1 , |b2 | |a2 + b2 |p−1 , . . .)
≤ {Φ(|a1 |p , |a2 |p , . . .)1/p + Φ(|b1 |p , |b2 |p , . . .)1/p }
× Φ(|a1 + b1 |p , |a2 + b2 |p , . . .)1/q
thanks to (1). Hence Φ(p) (a + b) ≤ Φ(p) (a) + Φ(p) (b). The remaining properties of Φ(p) ,
being a symmetric gauge function as well as the last assertion are obvious.
(0)
P r o o f o f P r o p o s i t i o n 2.12. For any ε > 0, since B ∈ CΦ (H), there exists a
projection P of finite rank such that kBP ⊥ k < ε where P ⊥ = I − P . In view of Lemma
2.13(2) let k · k(2) denote the symmetric norm corresponding to Φ(2) . Then we get by
assumption
k |Aj |1/2 P ⊥ k(2) = kP ⊥ |Aj |P ⊥ k1/2 ≤ kP ⊥ BP ⊥ k1/2 < ε1/2
and similarly k |A∗j |1/2 P ⊥ k(2) < ε1/2 . Since Φ(ab) ≤ Φ(2) (a)Φ(2) (b) for a, b ∈ sfin by
Lemma 2.13(1), the generalized Hölder inequality (Lemma 2.8) gives
kAj P ⊥ k = k |Aj |P ⊥ k ≤ k |Aj |1/2 k(2) k |Aj |1/2 P ⊥ k(2) ≤ kBk1/2 ε1/2 ,
and similarly kP ⊥ Aj P k ≤ kA∗j P ⊥ k ≤ kBk1/2 ε1/2 . The same inequalities hold also for A.
Therefore
kAj − Ak ≤ k(Aj − A)P ⊥ k + kP ⊥ (Aj − A)P k + kP (Aj − A)P k
≤ 4kBk1/2 ε1/2 + kP (Aj − A)P k.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
143
Since P is of finite rank and Aj → A in WOT, it immediately follows that kP (Aj −
A)P k → 0. Since ε is arbitrary, we have kAj − Ak → 0.
For instance, if Aj , B ∈ B(H)+ , Aj ≤ B ∈ C(H), and Aj → A ∈ B(H) in WOT (hence
A ≤ B), then kAj − Ak∞ → 0. As is remarked in [79, p. 39], the condition |A∗j | ≤ B
cannot be dropped in the above proposition.
2.4. Ando’s extension of Birman–Koplienko–Solomyak majorization result. The final
result of this section is taken from [3]. (See [42, Appendix] by H. Kosaki for its extension
to the von Neumann algebra case.) Before stating the theorem, let us recall the notion
of operator monotone functions. A continuous real function f on [0, ∞) is said to be
operator monotone if A ≤ B implies f (A) ≤ f (B) for any A, B ∈ B(H)+ . Such a function
is characterized by the following integral representation:
(2.6)
f (x) = a + bx +
\ xt
∞
0
x+t
dν(t),
x ≥ 0,
T∞
where a ∈ R, b ≥ 0, and ν is a positive measure on (0, ∞) with 0 t(1 + t)−1 dν(t) < ∞.
This type of integral representation is a central result in the Löwner theory of operator
monotone functions (see [1, 24, 36] for details). The operator monotonicity of xθ on [0, ∞)
with 0 < θ < 1 is due to the Löwner–Heinz inequality. Also it is well known that log(1+x)
is operator monotone.
Theorem 2.14. Let A, B ∈ B(H)+ and k · k be any symmetric norm.
(1) If f is an operator monotone function on [0, ∞) with f (0) ≥ 0, then
(2.7)
µ(f (A) − f (B)) ≺w µ(f (|A − B|)).
(2) If g is a continuous strictly increasing function on [0, ∞) such that g(0) = 0 and
limt→∞ g(t) = ∞ and whose inverse function is operator monotone, then
θ
µ(g(|A − B|)) ≺w µ(g(A) − g(B)).
When f (x) = x with 0 < θ < 1, the weak majorization (2.7) above was formerly
proved by Birman, Koplienko, and Solomyak [18], which gives the generalized Powers–
Størmer inquality: If 0 < θ < 1 and θ ≤ p ≤ ∞, then
(2.8)
kAθ − B θ kp/θ ≤ kA − Bkθp ,
A, B ∈ B(H)+ .
The case when θ = 1/2 and p = 1 is known as the Powers–Størmer inequality [72], whose
extension to the von Neumann algebra setting was obtained in [9, 34].
We first prepare simple facts to prove the theorem.
Lemma 2.15. Let X, Y ∈ B(H) be self-adjoint and X = X+ − X− , Y = Y+ − Y− be
the Jordan decompositions.
(1) If X ≤ Y then µi (X+ ) ≤ µi (Y+ ) for all i.
(2) If µ(X+ ) ≺w µ(Y+ ) and µ(X− ) ≺w µ(Y− ), then µ(X) ≺w µ(Y ).
P r o o f. (1) Let Q be the support projection of X+ . Since
X+ = QXQ ≤ QY Q ≤ QY+ Q,
we have µi (X+ ) ≤ µi (QY+ Q) ≤ µi (Y+ ) by Proposition 1.4.
144
F. HIAI
(2) It is rather easy to see that µ(X) is the decreasing rearrangement of the combination of µ(X+ ) and µ(X− ). So for each k ∈ N we can choose 0 ≤ m ≤ k so that
k
X
µi (X) =
m
X
µi (X+ ) +
i=1
i=1
i=1
k−m
X
µi (X− ).
Hence
k
X
i=1
µi (X) ≤
m
X
µi (Y+ ) +
k−m
X
µi (Y− ) ≤
i=1
i=1
k
X
µi (Y ),
i=1
as desired.
P r o o f o f T h e o r e m 2.14. (1) First assume that A ≥ B ≥ 0 and let C = A−B ≥ 0.
In view of Proposition 2.10 it suffices to prove that
(2.9)
kf (B + C) − f (B)k(k) ≤ kf (C)k(k) ,
k ∈ N.
For t > 0 let
t
x
=1−
,
x+t
x+t
which is increasing on [0, ∞) with ht (0) = 0. According to the integral representation
(2.6) with a, b ≥ 0, we have
ht (x) =
µi (f (C)) = f (µi (C)) = a + bµi (C) +
\ µi (C)t
∞
0
µi (C) + t
so that
(2.10)
kf (C)k(k) ≥ bkCk(k) +
dν(t) = a + bµi (C) +
\
∞
tµi (ht (C))dν(t),
0
\
∞
tkht (C)k(k) dν(t).
0
On the other hand, since
f (B + C) = aI + b(B + C) +
\
∞
tht (B + C)dν(t)
0
as well as the analogous expression for f (B), we have
f (B + C) − f (B) = bC +
\
∞
0
t{ht (B + C) − ht (B)}dν(t),
so that
(2.11)
kf (B + C) − f (B)k(k) ≤ bkCk(k) +
\
∞
0
tkht (B + C) − ht (B)k(k) dν(t).
By (2.10) and (2.11) it suffices for (2.9) to show that
kht (B + C) − ht (B)k(k) ≤ kht (C)k(k) ,
t > 0, k ∈ N.
Since ht (x) = h1 (x/t), the case t = 1 is enough because we may replace B and C by
t−1 B and t−1 C, respectively. So what remains to prove is the following:
(2.12)
k(B + I)−1 − (B + C + I)−1 k(k) ≤ kI − (C + I)−1 k(k) ,
k ∈ N.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
145
Since
(B + I)−1 − (B + C + I)−1 = (B + I)−1/2 h1 ((B + I)−1/2 C(B + I)−1/2 )(B + I)−1/2
and k(B + I)−1/2 k∞ ≤ 1, we obtain
µi ((B + I)−1 − (B + C + I)−1 ) ≤ µi (h1 ((B + I)−1/2 C(B + I)−1/2 ))
= h1 (µi ((B + I)−1/2 C(B + I)−1/2 ))
≤ h1 (µi (C)) = µi (I − (C + I)−1 )
by repeated use of Proposition 1.4(7). Therefore (2.12) is proved.
Next let us prove the general case A, B ≥ 0. Since 0 ≤ A ≤ B + (A − B)+ , it follows
that
f (A) − f (B) ≤ f (B + (A − B)+ ) − f (B),
which implies by Lemma 2.15(1) that
k(f (A) − f (B))+ k(k) ≤ kf (B + (A − B)+ ) − f (B)k(k) .
Applying the first case to B + (A − B)+ and B, we get
kf (B + (A − B)+ ) − f (B)k(k) ≤ kf ((A − B)+ )k(k) .
Therefore
(2.13)
µ((f (A) − f (B))+ ) ≺w µ(f ((A − B)+ )).
Exchanging the role of A, B gives
(2.14)
µ((f (A) − f (B))− ) ≺w µ(f ((A − B)− )).
Here we may assume f (0) = 0 because f can be replaced by f −f (0). Then it is immediate
that f ((A − B)+ )f ((A − B)− ) = 0 and f ((A − B)+ ) + f ((A − B)− ) = f (|A − B|). So
µ(f (A)−f (B)) ≺w µ(f (|A−B|)) follows from (2.13) and (2.14) thanks to Lemma 2.15(2).
(2) Let f be the inverse of g. Since f satisfies the condition of (1), we have
k
X
i=1
µi (f (A) − f (B)) ≤
k
X
i=1
f (µi (A − B)),
k ∈ N.
Here replace A and B by g(A) and g(B), respectively. Then
k
X
i=1
µi (A − B) ≤
k
X
i=1
f (µi (g(A) − g(B))),
k ∈ N,
which means that µ(A − B) ≺w f (µ(g(A) − g(B))). As is well known, the operator
monotonicity of f implies the concavity of f , so that g is convex. Hence µ(g(|A − B|)) =
g(µ(A − B)) ≺w µ(g(A) − g(B)) by Proposition 1.2.
Problem 2.16. Let A, B ∈ B(H)+ and f be an operator monotone function on [0, ∞)
with f (0) = 0. Then it is natural to ask whether or not the following variant of (2.7)
holds:
µ(f (A + B)) ≺w µ(f (A) + f (B)).
146
F. HIAI
But it seems that the method in proving (2.7) does not work in this case. Theorem 2.14(1)
and (1.12) show that
µ(f (A + B)) ≺w µ(f (A)) + µ(f (B)),
while a stronger result is found in [4, Theorem 6.9].
3. Inequalites of Golden–Thompson type. This section is mostly taken from
[40]. We obtain log-majorization results and norm inequalities of Golden–Thompson type
for exponential operators. Furthermore, we discuss the norm convergence of exponential
product formula, which is the main result of [40]. But accounts more detailed than those
in [40] are supplied concerning technical parts: Lemmas 3.1, 3.2, 3.8, and theory of state
perturbation in von Neumann algebras.
3.1. Antisymmetric tensor powers. First let us establish a machinery of antisymmetric
tensors, which is quite useful in deriving log-majorization results. Let H be a separable
Hilbert space as before. For each n ∈ N let ⊗n H denote the n-fold tensor product of H
with itself, which is the completed Hilbert space of the n-fold algebraic tensor product
with respect to the inner product defined by
hξ1 ⊗ . . . ⊗ ξn , η1 ⊗ . . . ⊗ ηn i =
n
Y
i=1
hξi , ηi i.
For ξ1 , . . . , ξn ∈ H define ξ1 ∧ . . . ∧ ξn ∈ ⊗n H by
1 X
(sign π)ξπ(1) ⊗ . . . ⊗ ξπ(n) ,
(3.1)
ξ1 ∧ . . . ∧ ξn = √
n! π
where π runs over all permutations on {1, . . . , n} and sign π = ±1 according as π is
even or odd. The closed subspace of ⊗n H spanned by {ξ1 ∧ . . . ∧ ξn : ξi ∈ H} is called
the n-fold antisymmetric tensor product of H and denoted by Λn H. In fact, the linear
extension of the map ξ1 ⊗ . . . ⊗ ξn 7→ √1n! ξ1 ∧ . . . ∧ ξn is the projection of ⊗n H onto Λn H.
A straightforward computation from (3.1) shows that
(3.2)
hξ1 ∧ . . . ∧ ξn , η1 ∧ . . . ∧ ηn i = det[hξi , ηj i]1≤i,j≤n .
Note that ξ1 ∧ . . . ∧ ξn 6= 0 if and only if {ξ1 , . . . , ξn } is linearly independent. Moreover, if
{ϕi } is an orthonormal basis of H, then {ϕi1 ∧ . . . ∧ ϕin : i1 < . . . < in } is an orthonormal
basis of Λn H.
For every A ∈ B(H) the n-fold tensor product ⊗n A ∈ B(⊗n H) is given as
(⊗n A)(ξ1 ⊗ . . . ⊗ ξn ) = Aξ1 ⊗ . . . ⊗ Aξn .
Since Λn H is invariant for ⊗n A, the antisymmetric tensor power Λn A of A can be defined
as Λn A = ⊗n A|Λn H ; in fact,
(3.3)
(Λn A)(ξ1 ∧ . . . ∧ ξn ) = Aξ1 ∧ . . . ∧ Aξn .
When dim H = N < ∞, ΛN H = C, ΛN A = det A, and Λn H = {0} for n > N . In the
matrix theory Λn A is usually called the nth compound of A.
The following are elementary properties of antisymmetric tensor powers.
147
LOG-MAJORIZATIONS AND NORM INEQUALITIES
Lemma 3.1. Let X, Xj , Y, A ∈ B(H) and n ∈ N.
(1) Λn (X ∗ ) = (Λn X)∗ .
(2) Λn (XY ) = (Λn X)(Λn Y ).
(3) If kXj − Xk∞ → 0, then kΛn Xj − Λn Xk∞ → 0.
(4) If Xj → X in SOT (resp. WOT ) and supj kXj k∞ < ∞, then Λn Xj → Λn X in
SOT (resp. WOT ).
(5) If A ≥ 0, then Λn A ≥ 0 and Λn (Ap ) = (Λn A)p for all p > 0.
(6) Λn (|X|) = |Λn X|.
P r o o f. (1) and (2) are the restrictions of the corresponding formulas ⊗n (X ∗ ) =
(⊗ X)∗ and ⊗n (XY ) = (⊗n X)(⊗n Y ) to Λn H. For (3) and (4) it suffices to show the
corresponding convergences for ⊗n , which are readily verified. If A ≥ 0 then Λn A =
(Λn (A1/2 ))∗ (Λn (A1/2 )) ≥ 0 by (1) and (2). When p is rational, the second assertion of
(5) is immediate from (2). Then (3) implies the case of general p > 0. Finally (6) follows
from (1), (2), and (5).
n
A useful technique in the majorization theory for operators is supplied by the following
lemma. For matrices this is a consequence of the so-called Binet–Cauchy theorem ([62,
pp. 503–504]).
Lemma 3.2. For every A ∈ B(H) and n ∈ N,
n
Y
(3.4)
i=1
µi (A) = µ1 (Λn A) (= kΛn Ak∞ ).
P r o o f. We may assume by Lemma 3.1(6) that A ≥ 0. First suppose that A is
compact. Then there exists an orthonormal basis {ϕi } of H such that Aϕi = µi (A)ϕi for
all i. Since {ϕi1 ∧ . . . ∧ ϕin : i1 < . . . < in } is a complete set of eigenvectors of Λn A with
n
o
nY
µij (A) ϕi1 ∧ . . . ∧ ϕin ,
(Λn A)(ϕi1 ∧ . . . ∧ ϕin ) =
j=1
we have (3.4). Next let A ≥ 0 be general. Let Pk be projections of finite rank such that
Pk ↑ I. Since Pk APk → A in SOT, Lemma 3.1(4) implies that Λn (Pk APk ) → Λn A in
SOT. Since k · k∞ is lower-semicontinuous in WOT, we have
kΛn Ak∞ ≤ lim inf kΛn (Pk APk )k∞ = lim inf
k→∞
k→∞
n
Y
i=1
µi (Pk APk ) ≤
n
Y
µi (A)
i=1
by the first case and Proposition 1.4(7). On the other hand, by Lemma 1.16 we can choose
(k)
(k)
(k)
(k)
sequences of orthonormal sets {ξ1 , . . . , ξn } such that limk kAξi − µi (A)ξi k = 0 for
1 ≤ i ≤ n. Hence it follows from (3.2) and (3.3) that
(k)
(k)
(k)
(k)
kΛn Ak∞ ≥ h(Λn A)(ξ1 ∧. . .∧ξn(k) ), ξ1 ∧. . .∧ξn(k) i = det[hAξi , ξj i]1≤i,j≤n →
n
Y
µi (A)
i=1
as k → ∞, completing the proof.
Before going into the main part of this section, let us prove the Weyl majorization
theorem as warming-up practice in the antisymmetric tensor technique.
148
F. HIAI
Theorem 3.3. Let A ∈ C(H) and λ1 (A), λ2 (A), . . . be the eigenvalues of A arranged
as |λ1 (A)| ≥ |λ2 (A)| ≥ . . . with algebraic multiplicities counted. Then
n
Y
i=1
|λi (A)| ≤
n
Y
µi (A),
n ∈ N.
i=1
P r o o f. If λ is an eigenvalue of A with algebraic multiplicity m, then there exists a
set {η1 , . . . , ηm } of independent vectors such that
Aηj − ληj ∈ span{η1 , . . . , ηj−1 },
1 ≤ j ≤ m.
Hence for each n ∈ N we can choose independent vectors ξ1 , . . . , ξn such that Aξi =
λi (A)ξi + ζi with ζi ∈ span{ξ1 , . . . , ξi−1 } for 1 ≤ i ≤ n. Then it is readily checked that
n
o
nY
λi (A) ξ1 ∧ . . . ∧ ξn
(Λn A)(ξ1 ∧ . . . ∧ ξn ) = Aξ1 ∧ . . . ∧ Aξn =
i=1
and ξ1 ∧ . . . ∧ ξn 6= 0, so that
Qn
n
i=1 λi (A) is an eigenvalue of Λ A. Hence Lemma 3.2 gives
n
n
Y
Y
n
µi (A).
λi (A) ≤ kΛ Ak∞ =
i=1
i=1
3.2. Araki’s log-majorization result. Extending a trace inequality of Lieb and Thirring
[60], Araki [12] showed the following log-majorization result by essentially the same
method as below. Also, the same result for matrices was shown in [88] in the same way.
Note that this was further generalized in [55] to the von Neumann algebra case, while
the antisymmetric tensor technique can be no longer used. A generalization in another
direction will be given in Proposition 5.8.
Theorem 3.4. For every A, B ∈ B(H)+ ,
µ((A1/2 BA1/2 )r ) ≺w(log) µ(Ar/2 B r Ar/2 ),
(3.5)
r ≥ 1,
or equivalently
(3.6)
µ((Ap/2 B p Ap/2 )1/p ) ≺w(log) µ((Aq/2 B q Aq/2 )1/q ),
0 < p ≤ q.
P r o o f. We can pass to the limit from A + εI and B + εI as ε ↓ 0 by Proposition
1.4(10). So we may assume that A and B are invertible. First let us show that
(3.7)
k(A1/2 BA1/2 )r k∞ ≤ kAr/2 B r Ar/2 k∞ ,
r ≥ 1.
To do so, it suffices to show that Ar/2 B r Ar/2 ≤ I implies A1/2 BA1/2 ≤ I, equivalently
B r ≤ A−r implies B ≤ A−1 . But this is just the Löwner–Heinz inequality. For every
n ∈ N, since by Lemma 3.1,
Λn ((A1/2 BA1/2 )r ) = ((Λn A)1/2 (Λn B)(Λn A)1/2 )r ,
Λn (Ar/2 B r Ar/2 ) = (Λn A)r/2 (Λn B)r (Λn A)r/2 ,
it follows from (3.7) with Λn A, Λn B instead of A, B that
kΛn ((A1/2 BA1/2 )r )k∞ ≤ kΛn (Ar/2 B r Ar/2 )k∞ .
LOG-MAJORIZATIONS AND NORM INEQUALITIES
149
This means thanks to Lemma 3.2 that
n
n
Y
Y
µi (Ar/2 B r Ar/2 ).
µi ((A1/2 BA1/2 )r ) ≤
i=1
i=1
Hence (3.5) is proved. If we replace A, B by Ap , B p and take r = q/p, then
µ((Ap/2 B p Ap/2 )q/p ) ≺w(log) µ(Aq/2 B q Aq/2 ),
which implies (3.6) by Proposition 1.4(11).
Theorem 3.4 and Proposition 2.10 give:
Corollary 3.5. Let A, B ∈ B(H)+ and k · k be any symmetric norm. If f is a
continuous increasing function on [0, ∞) such that f (0) ≥ 0 and f (ex ) is convex , then
kf ((A1/2 BA1/2 )r )k ≤ kf (Ar/2 B r Ar/2 )k,
r ≥ 1.
In particular ,
k(A1/2 BA1/2 )r k ≤ kAr/2 B r Ar/2 k,
r ≥ 1.
3.3. Trotter–Kato exponential product formula. To obtain norm inequalities of Golden
–Thompson type, let us recall the form sum of positive self-adjoint operators. Let H
and K be positive self-adjoint (not necessarily bounded) operators on H. Let D0 =
D(H 1/2 ) ∩ D(K 1/2 ), D(H 1/2 ) being the domain of H 1/2 , H0 the closure of D0 , and P0
the projection onto H0 . We denote by H +̂ K the form sum of H and K, which is the
positive self-adjoint operator on H0 associated with the closed positive quadratic form
ξ ∈ D0 7→ kH 1/2 ξk2 + kK 1/2 ξk2 so that
k(H +̂K)1/2 ξk2 = kH 1/2 ξk2 + kK 1/2 ξk2 ,
ξ ∈ D0 .
The form sum H +̂K can be defined also for lower-bounded self-adjoint H, K as follows:
H +̂K = (H+ +̂K+ ) − P0 (H− + K− )P0 ,
where H = H+ − H− is the Jordan decomposition; or equivalently
H +̂K = ((H + aI)+̂(K + bI)) − (a + b)P0
by taking a, b ∈ R such that H + aI, K + bI ≥ 0. Note that if H + K is essentially
self-adjoint, then H +̂K coincides with the closure of H + K.
The following (1) is the so-called Trotter–Kato product formula, which was shown in
[51] in a more general form (see also [19]). The symmetric and continuous parameter version (2) was given in [40]. It is a simpler fact that if X, Y ∈ B(H) then limn→∞ (eX/n eY /n )n
= eX+Y in the norm k · k∞ . But a quite general Trotter-like convergence for exponential
products of bounded operators was established in [82].
Theorem 3.6. (1) If H and K are lower-bounded self-adjoint operators on H, then
s-lim(e−tH/n e−tK/n )n = e−t(H +̂K) ,
n→∞
t > 0,
the convergence being uniform in t ∈ [a, b] for any 0 < a < b. Here s-lim means the
convergence in SOT and e−t(H +̂K) is written for e−t(H +̂K) P0 for brevity.
150
F. HIAI
(2) With the same assumption and notation,
s-lim(e−rtH/2 e−rtK e−rtH/2 )1/r = e−t(H +̂K) ,
r↓0
t > 0,
the convergence being uniform in t ∈ [a, b] for any 0 < a < b.
P r o o f. Let us only show (2), which is an easy consequence of (1). We may assume
H, K ≥ 0 by taking H + aI, K + aI. For 0 < r < 1 write 1/r = n + s where n = n(r) ∈ N
and s = s(r) ∈ [0, 1). Then
(e−rtH/2 e−rtK e−rtH/2 )1/r
= (e−rtH/2 e−rtK e−rtH/2 )s e−rtH/2 e−rtK (e−rtH e−rtK )n−1 e−rtH/2 .
When r ↓ 0 (hence n → ∞), we have e−rtH/2 → I and e−rtK → I in SOT uniformly in
t ∈ [0, b]. Since
e−rtH/2 e−rtK e−rtH/2 ≤ (e−rtH/2 e−rtK e−rtH/2 )s ≤ I,
we get
s-lim(e−rtH/2 e−rtK e−rtH/2 )s = I
(3.8)
r↓0
uniformly in t ∈ [0, b]. Since r(n − 1)t → t, the uniform convergence of (1) implies that
(3.9)
s-lim{(e−rtH e−rtK )n−1 − e−r(n−1)t(H +̂K) } = 0
r↓0
uniformly in t ∈ [a, b]. Finally, it is immediate that
(3.10)
s-lim{e−r(n−1)t(H +̂K) − e−t(H +̂K) } = 0
r↓0
uniformly in t ∈ [0, b]. The above uniform convergences (3.8)–(3.10) altogether yield the
conclusion.
3.4. Log-majorization and norm inequalities of Golden–Thompson type. The next theorem is the Golden–Thompson inequality strengthened to the form of log-majorization.
Theorem 3.7. If H and K are lower-bounded self-adjoint operator on H, then
µ(e−(H +̂K) ) ≺w(log) µ((e−rH/2 e−rK e−rH/2 )1/r ),
r > 0.
To prove the theorem, we give the following infinite-dimensional extension of Ky Fan’s
multiplicative formula.
n
Y
i=1
Lemma 3.8. For every A ∈ B(H) and n ∈ N,
µi (A) = sup{det(P |A|P |P H ) : P is a projection of rank n}
= sup{Re det(P U AP |P H ) : U is a unitary and P is a projection of rank n}.
Qn
Hence the function A 7→ i=1 µi (A) is lower-semicontinuous in WOT on B(H).
P r o o f. If P is a projection of rank n and U is a unitary, then
n
n
Y
Y
µi (A),
µi (P |A|P ) ≤
det(P |A|P |P H ) =
i=1
i=1
151
LOG-MAJORIZATIONS AND NORM INEQUALITIES
Re det(P U AP |P H ) ≤ | det(P U AP |P H )| =
n
Y
i=1
µi (P U AP ) ≤
n
Y
µi (A).
i=1
To show the converse, we may assume that µn (A) > 0. By Lemma 1.16 (and its proof),
(k)
(k)
we can choose sequences of orthonormal sets {ξ1 , . . . , ξn } in the range of |A| such
(k)
(k)
that limk k |A|ξi − µi (A)ξi k = 0 for 1 ≤ i ≤ n. Let Pk be the projection onto
(k)
(k)
the span of {ξ1 , . . . , ξn } and A = W |A| the polar decomposition. For each k, since
(k)
(k)
(k)
(k)
{W ξ1 , . . . , W ξn } is orthonormal, there exists a unitary Uk such that Uk ξi = W ξi
for 1 ≤ i ≤ n. Then
(k)
(k)
det(Pk |A|Pk |Pk H ) = det[h|A|ξi , ξj i]i,j ,
(k)
(k)
(k)
(k)
(k)
(k)
det(Pk Uk∗ APk |Pk H ) = det[hAξi , Uk ξj i] = det[hAξi , W ξj i] = det[h|A|ξi , ξj i].
Q
(k) (k)
Since det[h|A|ξi , ξj i]ij → ni=1 µi (A) as k → ∞, we get the required formulas. Furthermore, the last assertion is immediate because A 7→ Re det(P U AP |P H ) is continuous
in WOT whenever P is of finite rank.
By the way, the following additive formula can be shown in a similar way: For every
A ∈ B(H) and n ∈ N,
n
X
i=1
µi (A) = sup{tr(P |A|P ) : P is a projection of rank n}
= sup{Re tr(P U AP ) : U is a unitary and P is a projection of rank n}
Pn
and hence i=1 µi (A) is lower-semicontinuous in WOT. However, we do not know an
explicit counter-example to µn (A) itself being lower-semicontinuous in WOT.
Indeed, Lemmas 3.1(4) and 3.2 serve for the proof of Theorem 3.7. But the above
multiplicative and additive formulas are worth pointing out by themselves.
P r o o f o f T h e o r e m 3.7. For every n ∈ N we have
n
Y
i=1
µi (e−(H +̂K) ) ≤ lim inf
r↓0
n
Y
i=1
by Theorem 3.6(2) and Lemma 3.8. Since
r ↓ 0 by (3.6), we conclude that
n
Y
i=1
µi (e−(H +̂K) ) ≤
n
Y
µi ((e−rH/2 e−rK e−rH/2 )1/r )
Qn
i=1
µi ((e−rH/2 e−rK e−rH/2 )1/r ) decreases as
µi ((e−rH/2 e−rK e−rH/2 )1/r ),
r > 0,
i=1
as desired.
Corollary 3.9. If H and K are lower-bounded self-adjoint operators on H and k · k
is a symmetric norm, then
ke−(H +̂K) k ≤ k(e−rH/2 e−rK e−rH/2 )1/r k,
and the above right-hand side decreases as r ↓ 0. In particular ,
(3.11)
r > 0,
ke−(H +̂K) k ≤ ke−H/2 e−K e−H/2 k ≤ ke−H e−K k.
152
F. HIAI
P r o o f. The first assertion is a consequence of Theorem 3.7, (3.6), and Proposition
2.10. The second inequality of (3.11) follows because by Proposition 1.4(5)
ke−H e−K k = k |e−K e−H | k = k(e−H e−2K e−H )1/2 k.
The specialization of (3.11) to the trace norm k·k1 is the celebrated Golden–Thompson
trace inequality independently established in [33, 83, 84]. It was shown in [81] that
tr eH+K ≤ tr(eH/n eK/n )n for every Hermitian matrices H, K and n ∈ N. For the matrix
case, (3.11) was given in [58, 85]. Also (3.11) for the norm k · k∞ is known as Segal’s
inequality ([73, p. 260]). A rather trivial consequence of (3.11) is that ke−(H +̂K) k ≤
ke−H k∞ ke−K k for any symmetric norm k · k. Hence, if Φ is a symmetric gauge function
and e−K ∈ CΦ (H), then e−(H +̂K) ∈ CΦ (H) for every lower-bounded H.
Concerning (quasi-)norms k · kp we have:
Corollary 3.10. (1) If H, K are as above and 0 < p ≤ ∞, then
ke−(H +̂K) kp ≤ k(e−rH/2 e−rK e−rH/2 )1/r kp ,
r > 0,
and the above the right-hand side decreases as r ↓ 0.
(2) When 0 < p, p1 , p2 ≤ ∞ and 1/p = 1/p1 + 1/p2 , if e−H ∈ Cp1 (H) and e−K ∈
Cp2 (H), then e−(H +̂K) ∈ Cp (H).
P r o o f. (1) follows from Corollary 3.9 because for any 0 < p < ∞,
1/p
k(e−rH/2 e−rK e−rH/2 )1/r kp = k(e−rH/2 e−rK e−rH/2 )p/r k1
1/p
and ke−(H +̂K) kp = ke−p(H +̂K) k1 . (2) is immediate from (3.11) and the Hölder inequality (Proposition 2.7).
The next example is given to show that k(e−rH/2 e−rK e−rH/2 )1/r k does not converge
to ke−(H +̂K) k as r ↓ 0 in general.
Example 3.11. Define 2 × 2 matrices P0 and Qn , n ∈ N, by
p
1 − 1/n
1 0
p
.
P0 =
, Qn = ξn ⊗ ξn with ξn =
0 0
1/n
Then P0 and Qn are projections with P0 ∧ Qn = 0, so that P0⊥ + Q⊥
n is invertible. Let
0 < δ < 1 be arbitrary. For each n, since exp{(log ε)(P0⊥ + Q⊥
)}
→
0 as ε ↓ 0, we can
n
choose 0 < εn < 1 such that
k exp{(log εn )(P0⊥ + Q⊥
n )}k∞ ≤ δ.
L∞
Putting Hn = (− log εn )P0⊥ and Kn = (− log εn )Q⊥
H =
n , we define
n=1 Hn and
L∞ 2
L∞
K = n=1 Kn , which are positive self-adjoint operators on H = 1 C . Then
e−rH/2 e−rK e−rH/2 =
=
∞
M
(e−rHn /2 e−rKn e−rHn /2 )
n=1
∞
M
r
⊥ r/2
{(P0 + εn P0⊥ )r/2 (Qn + εn Q⊥
}.
n ) (P0 + εn P0 )
n=1
LOG-MAJORIZATIONS AND NORM INEQUALITIES
153
Since
r
⊥ r/2
(P0 + εn P0⊥ )r/2 (Qn + εn Q⊥
n ) (P0 + εn P0 )
≥ (P0 + εn P0⊥ )r/2 Qn (P0 + εn P0⊥ )r/2 = (P0 + εn P0⊥ )r/2 ξn ⊗ (P0 + εn P0⊥ )r/2 ξn ,
we have
1
εr
ke−rH/2 e−rK e−rH/2 k∞ ≥ sup k(P0 + εn P0⊥ )r/2 ξn k2 = sup 1 − + n ≥ 1,
n
n
n
n
so that k(e−rH/2 e−rK e−rH/2 )1/r k∞ ≥ 1 for all r > 0. On the other hand,
e−(H +̂K) =
∞
M
e−(Hn +Kn ) =
n=1
and hence ke
−(H +̂K)
∞
M
exp{(log εn )(P0⊥ + Q⊥
n )}
n=1
k∞ ≤ δ.
3.5. Trace norm convergence of exponential product formula. In this subsection let H
and K be lower-bounded self-adjoint operators such that H +K is essentially self-adjoint.
For simplicity we denote the closure of H + K by the same H + K.
The following trace norm convergence of exponential product formula was established
in [40].
Theorem 3.12. If e−K ∈ C1 (H), then
(3.12)
lim k(e−rH/2 e−rK e−rH/2 )1/r − e−(H+K) k1 = 0,
(3.13)
lim k(e−rK/2 e−rH e−rK/2 )1/r − e−(H+K) k1 = 0,
(3.14)
(3.15)
r↓0
r↓0
lim k(e−H/n e−K/n )n+1 − e−(H+K) k1 = 0,
n→∞
lim tr(e−H/n e−K/n )n = tr e−(H+K) .
n→∞
First note that all the operators in the theorem belong to C1 (H). For instance, we
have (e−rH/2 e−rK e−rH/2 )1/r ∈ C1 (H) for all r > 0, because with H ≥ aI,
µk ((e−rH/2 e−rK e−rH/2 )1/r ) = µk (e−rK/2 e−rH e−rK/2 )1/r
≤ µk (e−ra e−rK )1/r = e−a µk (e−K ).
The proof of the theorem in [40] is based on Araki’s Trotter product formula [8]
involving the modular operator and Donald’s perturbation theory [23] for positive normal
functionals on von Neumann algebras. So, for convenience, we first give a brief survey on
these materials.
Let M be a von Neumann algebra and (M, H, J, P) be a standard form ([9, 34]) of M.
For example, a standard form of M = B(H) is given as (B(H), C2 (H), ∗ , C2 (H)+ ), where
B(H) is represented on C2 (H) by left multiplication and C2 (H)+ is the set of positive
+
operators in C2 (H). Let M+
∗ denote the positive part of the predual M∗ , and ϕ ∈ M∗
be faithful. Then ϕ is written as ϕ = h · Φ, Φi with the cyclic and separating vector Φ ∈ P.
Let ∆Φ be the modular operator associated with Φ (or ϕ). See [80] for modular theory
154
F. HIAI
of von Neumann algebras. For each h ∈ Msa where Msa is the self-adjoint part of M,
Araki [7] defined the perturbed vector Φh by
(3.16)
h
Φ =
∞
X
\
t1
0
0
1/2
n
(−1)
n=0
dt1
\
\
tn−1
dt2 . . .
t
dtn ∆tΦn h∆Φn−1
−tn
h . . . ∆tΦ1 −t2 hΦ,
0
∆zΦ1 h∆zΦ2 h . . . ∆zΦn h
where Φ is in the domain of
for z = (z1 , . . . , zn ) ∈ Cn with Re z ∈
n
{(s1 , . . . , sn ) ∈ R : s1 , . . . , sn ≥ 0, s1 + . . . + sn ≤ 1/2} and the sum in (3.16) absolutely
converges. Then Φh is also a cyclic and separating vector in P. It is known [8] that Φ is
in the domain of exp 12 (log ∆Φ − h) and
log ∆Φ − h
Φh = exp
Φ.
2
Moreover, the perturbed functional ϕh is defined by
ϕh = h · Φh , Φh i.
(3.17)
Note that Φh and ϕh here mean Φ−h and ϕ−h in the notation of [7, 8].
The following is a version of the Trotter product formula given in [8, Remarks 1, 2].
Proposition 3.13. Let Φ be as above. Then for every h ∈ Msa ,
1/2n −h/2n n
Φh = lim (∆Φ
n→∞
e
1/2n n
) Φ = lim (e−h/2n ∆Φ
n→∞
) Φ
strongly.
The relative entropy for positive normal functionals on a von Neumann algebra was
first introduced in [87] in the semifinite case and was extended in [10, 11] to the general
case. For each ϕ, ψ ∈ M+
∗ let Φ and Ψ be the vector representatives in P so that ϕ =
h · Φ, Φi and ψ = h · Ψ, Ψi. Then Araki’s relative entropy S(ψ, ϕ) is defined by
S(ψ, ϕ) = −h(log ∆Φ,Ψ )Ψ, Ψi
if the support of ψ is dominated by that of ϕ, and S(ψ, ϕ) = ∞ otherwise, where ∆Φ,Ψ
is the relative modular operator ([11]). See [86] for another definition and [54, 68] for
properties of relative entropy.
Next let h be a lower-bounded
T∞ self-adjoint operator affiliated with M. So we have the
spectral decomposition h = a λdeh (λ), where {eh (λ)}λ≥a is a spectral resolution in M
T∞
with a ∈ R. For any ψ ∈ M+
∗ , ψ(h) can be defined as ψ(h) = a λdψ(eh (λ)). For each
ϕ ∈ M+
∗ define c(ϕ, h) ∈ [−∞, ∞) by
c(ϕ, h) = sup{−ψ(h) − S(ψ, ϕ) : ψ ∈ Σ∗ (M)},
where Σ∗ (M) denotes the set of normal states on M. According to [23, Theorem 3.1],
if c(ϕ, h) > −∞, that is, ψ(h) + S(ψ, ϕ) < ∞ for some ψ ∈ Σ∗ (M), then there exists a
unique ω ∈ Σ∗ (M) such that −ω(h) − S(ω, ϕ) = c(ϕ, h). This ω is denoted by [ϕh ], while
it was denoted in [23] by ϕh . More generally, h can be an extended-valued lower-bounded
operator (see [23]), but a self-adjoint h is enough for our purpose.
h
Note [70, Proposition 1] that when ϕ ∈ M+
∗ is faithful and h ∈ Msa , [ϕ ] coincides
h
h
h
with (3.17) up to a normalization constant; more precisely [ϕ ] = ϕ /ϕ (I) and ϕh (I) =
ec(ϕ,h) . So for any ϕ ∈ M+
∗ and any lower-bounded self-adjoint operator h affiliated with
LOG-MAJORIZATIONS AND NORM INEQUALITIES
155
M such that c(ϕ, h) > −∞, we can define the perturbed functional ϕh by
ϕh = ec(ϕ,h) [ϕh ].
(3.18)
Then h is called a relative Hamiltonian of ω = ϕh relative to ϕ.
The following convergence result was proved in [23, Proposition 3.15], which plays a
crucial role in the proof of Theorem 3.12.
Proposition 3.14. Let hn and h be lower-bounded self-adjoint operators affiliated with
+
M such that hn ↑ h in the sense that ψ(hn ) ↑ ψ(h) for all ψ ∈ M+
∗ . If ϕ ∈ M∗ satisfies
h
hn
hn
c(ϕ, h) > −∞, then c(ϕ, hn ) → c(ϕ, h), k[ϕ ] − [ϕ ]k → 0, and hence kϕ − ϕh k → 0.
Now, under the above preparations, let us sketch the proof of Theorem 3.12.
P r o o f o f T h e o r e m 3.12 (Sketch). The von Neumann algebra M = B(H) is
represented by left multiplication on the standard Hilbert space C2 (H). Set Φ = e−K/2 ∈
C2 (H)+ , which is a cyclic and separating vector for M and defines a faithful ϕ ∈ M+
∗
as ϕ = h · Φ, Φi = tr( · e−K ). We write ϕ = e−K under the usual identification B(H)∗ =
C1 (H). Note that
−it
−itK
∆it
XeitK ,
Φ X∆Φ = e
(3.19)
t ∈ R, X ∈ B(H),
because both sides represent the modular automorphism group associated with Φ.
We may assume that H, K ≥ 0. First assume that H is bounded. An argument of
analytic continuation using (3.19) yields
1/2n n
(e−H/2n ∆Φ
(3.20)
−H/2n −K/2n n
) Φ = (e−H/2n e−K/2n )n ,
−(H+K)/2
Since (e
e
) → e
show that ΦH = e−(H+K)/2 and
n ∈ N.
in SOT as n → ∞, Proposition 3.13 and (3.20)
lim k(e−H/2n e−K/2n )n − e−(H+K)/2 k2 = 0,
n→∞
which yields
lim tr(e−H/4n e−K/2n e−H/4n )2n = tr e−(H+K) .
n→∞
This together with Corollary 3.10(1) implies that
(3.21)
T∞
lim tr(e−rH/2 e−rK e−rH/2 )1/r = tr e−(H+K) .
Tn
r↓0
Next let H = a λdEH (λ) be lower-bounded and Hn = a λdEH (λ). Since c(ϕ, H) >
−∞ follows from D(H) ∩ D(K) 6= {0}, we can define the perturbed functional ϕH by
(3.18) as well as ϕHn . Then Proposition 3.14 shows that kϕHn − ϕH k → 0. We have
ϕHn = e−(Hn +K) because ΦHn = e−(Hn +K)/2 for bounded Hn . Furthermore, (I + Hn +
K)−1 ↓ (I + H + K)−1 and hence e−(Hn +K) → e−(H+K) in SOT (see the proofs of
Lemmas 4.15 and 4.8(1) below). Therefore ϕH = e−(H+K) and
lim ke−(Hn +K) − e−(H+K) k1 = 0.
(3.22)
n→∞
By (3.21) for bounded Hn and by (3.22) we get
lim tr(e−rH/2 e−rK e−rH/2 )1/r ≤ lim tr(e−rHn /2 e−rK e−rHn /2 )1/r
r↓0
r↓0
= tr e−(Hn +K) → tr e−(H+K) ,
156
F. HIAI
which implies (3.21) for lower-bounded H. Hence
lim k(e−rH/2 e−rK e−rH/2 )1/2r k2 = ke−(H+K)/2 k2 .
r↓0
Since it is easily seen from Theorem 3.6(2) that (e−rH/2 e−rK e−rH/2 )1/2r → e−(H+K)/2
weakly in C2 (H), we have
lim k(e−rH/2 e−rK e−rH/2 )1/2r − e−(H+K)/2 k2 = 0,
r↓0
which shows (3.12) by using the Hölder inequality. The proof of (3.13) is similar, and
(3.14) is easily shown from (3.12) thanks to
(3.23)
(e−H/n e−K/n )n+1 = e−H/2n (e−H/2n e−K/n e−H/2n )n e−H/2n e−K/n .
Finally (3.15) is immediate from (3.12).
Corollary 3.15. If e−K ∈ Cp (H) where 0 < p < ∞, then
lim k(e−rH/2 e−rK e−rH/2 )1/r − e−(H+K) kp = 0,
r↓0
lim k(e−rK/2 e−rH e−rK/2 )1/r − e−(H+K) kp = 0,
r↓0
lim k(e−H/n e−K/n )n+1 − e−(H+K) kp = 0.
(3.24)
n→∞
Furthermore, for every q > p,
lim k(e−H/n e−K/n )n − e−(H+K) kq = 0.
n→∞
P r o o f (Sketch). Choose k ∈ N such that 2k > 1/p. Applying (3.12) to pH, pK
instead of H, K we have
lim k(e−rH/2 e−rK e−rH/2 )p/r − e−p(H+K) k1 = 0,
r↓0
so that by (2.8) with θ = 1/2k p
k
lim k(e−rH/2 e−rK e−rH/2 )1/2
r↓0
r
k
− e−(H+K)/2 k2k p = 0.
Now, using the Hölder inequality repeatedly, we can obtain the first assertion. (A similar
argument in more detail will be supplied in the proof of Corollary 4.23.) The second is
similar and the third is shown by using (3.23). The last assertion is also shown by using
(2.8) and the Hölder inequality (see [40] for details).
Problems 3.16. (1) Is it possible to replace (e−H/n e−K/n )n+1 by (e−H/n e−K/n )n in
(3.14) and (3.24)?
(0)
(2) In view of Corollary 3.15, we are tempted to conjecture that if e−K ∈ CΦ (H)
then
lim k(e−rH/2 e−rK e−rH/2 )1/r − e−(H+K) k = 0,
r↓0
where Φ is a symmetric gauge function with the corresponding norm k · k.
4. Inequalites of complementary Golden–Thompson type. A log-majorization
result in terms of power operator means was obtained in [6] in the matrix case, which gives
rise to matrix norm inequalities considered as complementary to the Golden–Thompson
LOG-MAJORIZATIONS AND NORM INEQUALITIES
157
ones. The aim of this section is to study log-majorizations and norm inequalities involving
operator means for infinite-dimensional exponential operators. In the course of this study,
we establish a Trotter-like exponential product formula for operator means, which may
be interesting by itself. Most results in this section are new.
4.1. Preliminaries on operator means. First, for convenience, let us give a brief survey
on operator means. An axiomatic approach for operator means was investigated by Kubo
and Ando [57]. Assume that H is a separable and infinite-dimensional Hilbert space. A
binary operation σ : B(H)+ × B(H)+ → B(H)+ is called an operator connection if it
satisfies the following conditions (i)–(iii) for A, B, C, D ∈ B(H)+ :
(i) A ≤ C and B ≤ D imply A σ B ≤ C σ D (joint monotonicity),
(ii) C(A σ B)C ≤ (CAC) σ (CBC) (transformer inequality),
(iii) An , Bn ∈ B(H)+ , An ↓ A, and Bn ↓ B imply An σ Bn ↓ A σ B (upper semicontinuity).
An operator connection σ is called an operator mean if
(iv) I σ I = I.
The fundamental theorem of Kubo and Ando is summarized as follows: For each
operator connection σ there exists a unique operator monotone function f ≥ 0 on [0, ∞)
such that f (x)I = I σ (xI) for x ≥ 0. Then the map σ 7→ f is an affine order-isomorphism
between the operator connections and the non-negative operator monotone functions on
[0, ∞). The order preservation means that when σi 7→ fi for i = 1, 2, A σ1 B ≤ A σ2 B for
all A, B ∈ B(H)+ if and only if f1 (x) ≤ f2 (x) for all x ≥ 0. The operator connection σ is
defined via the corresponding function f by
(4.1)
A σ B = A1/2 f (A−1/2 BA−1/2 )A1/2
if A is invertible, and for general A, B
−1/2
A σ B = s-lim A1/2
Bε Aε−1/2 )A1/2
ε f (Aε
ε
ε↓0
decreasingly,
where Aε = A + εI and Bε = B + εI. Moreover σ is an operator mean if and only if
f (1) = 1, which implies that A σ A = A for all A.
The following are typical examples of operator means.
(1) Arithmetic mean: A ∇ B = 12 (A + B).
−1 −1
(2) Harmonic mean: A ! B = 2(A : B), where A : B = s-limε↓0 (A−1
is
ε + Bε )
called the parallel sum of A, B.
−1/2 1/2 1/2
−1/2
1/2
) Aε .
Bε Aε
(3) Geometric mean: A # B = s-limε↓0 Aε (Aε
The corresponding functions of ∇, ! , # are (1 + x)/2, 2x/(x + 1), x1/2 , respectively.
For 0 ≤ α ≤ 1 let #α denote the α-power mean, which is the operator mean corresponding
to the operator monotone function xα . Namely, for each A, B ∈ B(H)+ with A invertible,
A #α B is defined by
A #α B = A1/2 (A−1/2 BA−1/2 )α A1/2 .
Here we adopt the usual convention B 0 = I for any B ∈ B(H)+ . Note that A #0 B = A,
A #1 B = B, and A #1/2 B = A # B.
158
F. HIAI
Using the integral representation (2.6) of f and transforming the measure ν to m =
t(1 + t)−1 ν, we can represent A σ B as
\ 1+t
\ 1+t
A σ B = aA + bB +
{(tA) : B}dm(t) =
{(tA) : B}dm(t),
t
t
(0,∞)
[0,∞]
where a = m({0}) and b = m({∞}). In this way, it is seen that there exists an affine
one-to-one correspondence between the operator means and the probability measures on
[0, ∞]. The above representation shows that every operator connection σ satisfies
(v) (A σ B) + (C σ D) ≤ (A + C) σ (B + D),
(vi) T ∗ (A σ B)T ≤ (T ∗ AT ) σ (T ∗ BT ) for every T ∈ B(H) and the equality holds if T
is invertible.
Now let σ be an operator mean with the corresponding function f . The integral
representation of f implies that f is automatically infinitely many times differentiable.
So let α = f ′ (1). The concavity of f implies that 0 ≤ α ≤ 1 and f (x) ≤ (1 − α) + αx for
x ≥ 0. Since f (x−1 )−1 is again operator monotone and hence concave, we have
x
≤ f (x) ≤ (1 − α) + αx,
x ≥ 0.
(4.2)
(1 − α)x + α
The operator mean σ ′ determined by A σ ′ B = B σ A is called the transpose of σ. The
corresponding function of σ ′ is xf (x−1 ) because
I σ ′ (xI) = (xI) σ I = x(I σ (x−1 I)) = xf (x−1 )I,
x > 0.
In particular, if σ is symmetric, i.e. A σ B = B σ A for all A, B, or equivalently f (x) =
xf (x−1 ), then α = 1/2 and so (4.2) says that the arithmetic mean A ∇ B is the greatest
and the harmonic mean A ! B is the least among the symmetric operator means. If α = 0,
then f (x) ≡ 1 and A σ B = A. Also if α = 1, then f (x) = x and A σ B = B. All the
results in this section become trivial in these two extremal cases. So we may assume
0 < α < 1 in the sequel discussions.
Although the next lemma will be used only for α-power mean #α , we give it for
general σ.
Lemma 4.1. Let σ be an operator mean and f the corresponding operator monotone
function. Let A, B ∈ B(H)+ .
(1) Assume that A is invertible. Then B 7→ A σ B is continuous in the norm k · k∞
on B(H)+ .
(2) Assume that f (0) = 0 and B is compact. If An ↓ A then kAn σ B − A σ Bk∞ → 0.
P r o o f. (1) is immediate from the expression (4.1) because of the k · k∞ -continuity
of functional calculus.
(2) Choose a > 0 such that An ≤ aI. Then we have An σ B ≤ (aI) σ B = af (a−1 B)
and af (a−1 B) ∈ C(H) from f (0) = 0. Hence Proposition 2.12 shows the assertion because
An σ B ↓ A σ B.
4.2. Log-majorization for power operator means. The log-majorization in the next
theorem was given in [6] in the matrix case, which is considered as complementary to
Theorem 3.4.
159
LOG-MAJORIZATIONS AND NORM INEQUALITIES
Theorem 4.2. If A, B ∈ B(H)+ and A is either invertible or compact , then
µ(Ar #α B r ) ≺w(log) µ((A #α B)r ),
(4.3)
r ≥ 1,
or equivalently
µ((Ap #α B p )1/p ) ≺w(log) µ((Aq #α B q )1/q ),
(4.4)
p ≥ q > 0.
P r o o f. First assume that both A and B are invertible. The proof below is the same
as that of [6, Theorem 2.1]. For each n ∈ N it is easily checked from Lemma 3.1 that
Λn (Ar #α B r ) = (Λn A)r #α (Λn B)r ,
Λn ((A #α B)r ) = ((Λn A) #α (Λn B))r .
So it suffices to show that
kAr #α B r k∞ ≤ k(A#α B)r k∞ ,
(4.5)
n
n
r ≥ 1,
because (4.3) follows from Lemma 3.2 by taking Λ A, Λ B instead of A, B in (4.5). To
show (4.5), we may prove that A #α B ≤ I implies Ar #α B r ≤ I. When 1 ≤ r ≤ 2, let
us write r = 2 − ε with 0 ≤ ε ≤ 1. Let C = A−1/2 BA−1/2 . Suppose A #α B ≤ I. Then
C α ≤ A−1 and
A ≤ C −α ,
(4.6)
so that thanks to 0 ≤ ε ≤ 1,
A1−ε ≤ C −α(1−ε) .
(4.7)
Now we have
ε
ε
ε
ε
Ar #α B r = A1− 2 {A−1+ 2 B · B −ε · BA−1+ 2 }α A1− 2
ε
= A1− 2 {A−
1−ε
2
CA1/2 (A−1/2 C −1 A−1/2 )ε A1/2 CA−
1−ε
2
= A1/2 {A1−ε #α [C(A #ε C −1 )C]}A1/2
ε
}α A1− 2
≤ A1/2 {C −α(1−ε) #α [C(C −α #ε C −1 )C]}A1/2
by using (4.6), (4.7), and the joint monotonicity of power means. Since
C −α(1−ε) #α [C(C −α #ε C −1 )C] = C −α(1−ε)(1−α) [C(C −α(1−ε) C −ε )C]α = C α ,
we get
(4.8)
Ar #α B r ≤ A1/2 C α A1/2 = A #α B ≤ I.
Therefore (4.3) is proved when 1 ≤ r ≤ 2. When r > 2, write r = 2k s with k ∈ N and
1 ≤ s ≤ 2. Repeating the above argument we have
k−1
µ(Ar #α B r ) ≺w(log) µ(A2
..
.
s
k−1
#α B 2
s 2
)
k
≺w(log) µ(As #α B s )2
≺w(log) µ(A #α B)r .
Second assume that A is invertible while B is general. Since by Lemma 4.1(1)
Ar #α B r = lim Ar #α Bεr and (A #α B)r = lim(A #α Bε )r
ε↓0
ε↓0
160
F. HIAI
in the norm k·k∞ , (4.3) follows from the first case and Proposition 1.4(10). Third assume
that A is compact. The second case shows that
µ(Bεr #1−α Ar ) ≺w(log) µ((Bε #1−α A)r ),
r ≥ 1, ε > 0.
Since Lemma 4.1(2) implies that Bε #1−α A → B #1−α A and Bεr #1−α Ar → B r #1−α Ar
in k · k∞ , we get
µ(B r #1−α Ar ) ≺w(log) µ((B #1−α A)r ),
r ≥ 1,
which is nothing but (4.3). Finally (4.4) readily follows from (4.3) as in the last part of
proof of Theorem 3.4.
By Theorem 4.2 and Proposition 2.10 we have:
Corollary 4.3. Let A, B ∈ B(H)+ and assume that A is either invertible or compact.
Let k · k be any symmetric norm. If f is a continuous increasing function on [0, ∞) such
that f (0) ≥ 0 and f (ex ) is convex , then
kf (Ar #α B r )k ≤ kf ((A #α B)r )k,
r ≥ 1.
In particular ,
kAr #α B r k ≤ k(A #α B)r k,
r ≥ 1.
Problem 4.4. It is desirable to prove Theorem 4.2 (hence Corollary 4.3) without
the assumption of A being invertible or compact. We can do so if the convergence
kA #α Bk∞ = limε↓0 k(A + εI) #α Bk∞ holds true for general A, B ∈ B(H)+ . However,
this is not the case as the following example shows.
Example 4.5. Let us consider the infinite direct sum of 2 × 2 matrices. Let P0 , ξn ,
L∞
L∞
and Qn be as in Example 3.11. Define P = n=1 P0 and Q = n=1 Qn , so that
P #Q =
∞
M
(P0 # Qn ),
n=1
∞
M
1 0
(P + εI) # Q =
((P0 + εI2 ) # Qn ) where I2 =
.
0 1
n=1
We have
(P0 + εI2 ) # Qn
= (P0 + εI2 )1/2 {(P0 + εI2 )−1/2 ξn ⊗ (P0 + εI2 )−1/2 ξn }1/2 (P0 + εI2 )1/2
= (P0 + εI2 )1/2 k(P0 + εI2 )−1/2 ξn k2
1/2
(P0 + εI2 )−1/2 ξn
(P0 + εI2 )−1/2 ξn
⊗
(P0 + εI2 )1/2
k(P0 + εI2 )−1/2 ξn k k(P0 + εI2 )−1/2 ξn k
ξn
ξn
= k(P0 + εI2 )−1/2 ξn k
⊗
k(P0 + εI2 )−1/2 ξn k k(P0 + εI2 )−1/2 ξn k
= k(P0 + εI2 )−1/2 ξn k−1 ξn ⊗ ξn .
161
LOG-MAJORIZATIONS AND NORM INEQUALITIES
Since
q
n−1
n(1+ε)
(P0 + εI2 )−1/2 ξn =  q
we get
k(P0 + εI2 ) # Qn k∞ =
1
nε

,
1
n−1
+
n(1 + ε) nε
−1/2
.
Letting ε ↓ 0 yields kP0 # Qn k∞ = 0 (this follows also from P0 ∧Qn = 0; see [57, Theorem
3.7]). Hence P # Q = 0. On the other hand, we have
−1/2
1
n−1
= (1 + ε)1/2 .
k(P + εI) # Qk∞ = sup k(P0 + εI2 ) # Qn k∞ = sup
+
nε
n≥1
n≥1 n(1 + ε)
Therefore
kP # Qk∞ = 0 < 1 = lim k(P + εI) # Qk∞ .
ε↓0
Furthermore, for each 0 < δ < 1 and n ∈ N, we can choose αn > 0 such that
k(P0 + αn I2 ) # (Qn + αn I2 )k∞ ≤ δ.
L∞
Define A = n=1 (P0 + αn I2 ) and B = n=1 (Qn + αn I2 ), which are strictly positive.
Since P ≤ A and Q ≤ B, we have
L∞
kA # Bk∞ ≤ δ < 1 ≤ lim k(A + εI) # Bk∞ .
ε↓0
The next proposition extends [2, Theorem 1] where H, K are bounded and α = 1/2.
Proposition 4.6. If H ∈ B(H) is self-adjoint and K is a lower-bounded self-adjoint
operator on H, then the following conditions are equivalent :
(i) (1 − α)H + αK ≥ 0;
(ii) e−tH #α e−tK ≤ I for all t ≥ 0;
(iii) t 7→ e−tH #α e−tK is a decreasing function from [0, ∞) to B(H)+ .
P r o o f. (i)⇒(ii). Suppose that (1−α)H +αK ≥ 0, i.e. K ≥ −α−1 (1−α)H. To prove
(ii), it suffices to show that
lim k(e−rH #α e−rK )1/r k∞ ≤ 1,
(4.9)
r↓0
1/r
ke−rH #α e−rK k∞ = k(e−rH #α e−rK )1/r k∞ increases as r ↓
α{(1 − α)kHk∞ }−1 , since I + rK ≥ I − α−1 (1 − α)rH ≥ 0 and
because
0<r<
is invertible, we get
e−rK ≤ (I + rK)−1 ≤
0 by (4.4). When
I − α−1 (1 − α)rH
−1
1−α
rH
I−
α
as well as e−rH ≤ (I + rH)−1 for small r > 0. Therefore
−1
−α
1−α
1−α
−(1−α)
−rH
−rK
−1
I−
rH
rH
= (I + rH)
,
e
#α e
≤ (I + rH) #α I −
α
α
162
F. HIAI
so that for small r > 0
k(e
−rH
#α e
−rK 1/r
)
Since
−(1−α)/r
lim(1 + rλ)
r↓0
k∞
−α/r 1−α
−(1−α)/r
.
I−
≤ (I + rH)
rH
α
∞
−α/r
1−α
1−α
1−
rλ
= (eλ )−(1−α) (e− α λ )−α = 1
α
uniformly in λ ∈ [−kHk∞ , kHk∞ ], it follows that
−α/r
1−α
−(1−α)/r
= 0,
rH
−
I
I
−
(I
+
rH)
lim r↓0
α
∞
implying (4.9).
(ii)⇒(iii). What we have to prove is that if A, B ∈ B(H)+ and A is invertible, then
A #α B ≤ I implies Ar #α B r ≤ A #α B for every r ≥ 1. When both A and B are
invertible, the result was shown in the proof of Theorem 4.2 (see (4.8)). When A is
invertible but B is not, let Bε = B + εI for ε > 0. Since
A #α (kA #α Bε k−1/α
Bε ) = kA #α Bε k−1
∞
∞ (A #α Bε ) ≤ I,
−1/α
the first case applied to A and kA #α Bε k∞
r
A
#α Bεr
Bε shows that for r ≥ 1
≤ kA #α Bε kr−1
∞ (A #α Bε ).
Letting ε ↓ 0 we have by Lemma 4.1(1)
Ar #α B r ≤ kA #α Bkr−1
∞ (A #α B) ≤ A #α B,
as desired.
(iii)⇒(i). For every ε > 0 and t > 0, it follows from (4.2) that
e−tH #α (e−tK + εI) = e−tH/2 {etH/2 (e−tK + εI)etH/2 }α e−tH/2
≥ e−tH/2 {(1 − α)I + αe−tH/2 (e−tK + εI)−1 e−tH/2 }−1 e−tH/2
≥ e−tH/2 {(1 − α)I + αe−tH/2 etK e−tH/2 }−1 e−tH/2
= {(1 − α)etH + αetK }−1 ,
which shows that e−tH #α e−tK ≥ {(1 − α)etH + αetK }−1 . Hence (iii) implies that (1−
α)etH + αetK ≥ I for every t > 0. So for every n ∈ N we have
(4.10)
αn(eK/n − I) ≥ −(1 − α)n(eH/n − I).
T∞
Let K = b λdEK (λ) be the spectral decomposition. For any ξ ∈ EK (c)H with b < c <
∞, (4.10) says that
c
\
α n(eλ/n − 1)dkEK (λ)ξk2 ≥ −(1 − α)hn(eH/n − I)ξ, ξi.
b
Tc
The above left-hand side tends as n → ∞ to α b λdkEK (λ)ξk2 = αhKξ, ξi, while the
right-hand side tends to −(1 − α)hHξ, ξi. Therefore h((1 − α)H + αK)ξ, ξi ≥ 0. This
S
implies (i) because c>b EK (c)H is a core of (1 − α)H + αK.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
163
R e m a r k 4.7. By using the Furuta inequality [28], a more general result for positive
invertible operators was shown in [29] (also [27]) as follows: If A, B ∈ B(H)+ are invertible,
the following conditions, among others, are equivalent:
(I) log A ≥ log B;
r
(II) Ar ≥ (Ar/2 B p Ar/2 ) p+r for all p, r ≥ 0;
t+2r
(III) for any t ≥ 0, A−r (Ar B p Ar ) p+2r A−r is a decreasing function of both p ≥ t and
1−α
r ≥ 0 to B(H)+ . If we set α = r/(p+r), then the above (II) is written as A−r #α B α r ≤
I for all r ≥ 0 and 0 ≤ α ≤ 1. Thus it is seen that (I) ⇔ (II) is nothing but (i) ⇔ (ii) of
Proposition 4.6.
To obtain the complementary counterpart of Theorem 3.7, let us introduce the additive version of antisymmetric tensor powers. For a lower-bounded self-adjoint operator
H on H and n ∈ N, we define the lower-bounded self-adjoint operator Σ n H on Λn H by
n
X
Σ nH =
{(⊗m−1 I) ⊗ H ⊗ (⊗n−m I)} n .
Λ H
m=1
More precisely, for any A ∈ B(H), since it is easily checked that
n
X
(4.11)
(⊗m−1 I) ⊗ A ⊗ (⊗n−m I)
m=1
commutes with the projection
of ⊗n H onto Λn H, we can define Σ n A as the T
restriction of
T
k
∞
n
(4.11) to Λ H. Let Hk = a λdEH (λ) with the spectral decomposition H = a λdEH (λ).
Then Σ n H is the limit of an increasing sequence {Σ n Hk }∞
k=1 of (mutually commuting)
self-adjoint operators in B(Λn H).
We prove (2) below as stated, while it will be used in the obvious case when H is
bounded.
Lemma 4.8. Let H and K be lower-bounded self-adjoint operators on H. For n ∈ N
define Σ n H and Σ n K as above.
n
(1) Λn (e−H ) = e−Σ H .
(2) Σ n (H +̂K) = (Σ n H)+̂(Σ n K).
P r o o f. We may assume H, K ≥ 0 by taking H + aI, K + aI.
(1) If H is bounded, then we have
n
Y
n
e−Σ H =
{(⊗m−1 I) ⊗ e−H ⊗ (⊗n−m I)} n = ⊗n e−H |Λn H = Λn (e−H ).
m=1
Tk
Λ H
For general H ≥ 0 let Hk = 0 λdEH (λ). Then I ≥ (I + Hk )−1 ↓ (I + H)−1 as k → ∞,
which implies that e−Hk → e−H in SOT, because e−H is the functional calculus of
(I + H)−1 for f (x) = exp(1 − x−1 ) on [0, 1]. Hence Λn (e−Hk ) → Λn (e−H ) in SOT by
n
n
Lemma 3.1(4). Similarly e−Σ Hk → e−Σ H in SOT, showing the assertion.
(2) Let P0 be the projection of H onto the closure H0 of D(H 1/2 ) ∩ D(K 1/2 ). Then
Σ n (H +̂K) is a positive self-adjoint operator on Λn H0 (⊂ Λn H). Since Λn P0 is the
n
projection of Λn H onto Λn H0 and (1) says that e−Σ (H +̂K) = Λn (e−(H +̂K) ) on Λn H0 ,
we have
n
e−Σ (H +̂K) · Λn P0 = Λn (e−(H +̂K) ),
164
F. HIAI
where e−(H +̂K) = e−(H +̂K) P0 by convention in Theorem 3.6. Hence using Theorem 3.6(1),
Lemma 3.1, and (1), we have
e−Σ
n
(H +̂K)
· Λn P0 = s-lim Λn ((e−H/k e−K/k )k ) = s-lim(Λn (e−H )1/k Λn (e−K )1/k )k
k→∞
= s-lim(e
k→∞
k→∞
−(Σ n H)/k −(Σ n K)/k k
) = e−((Σ
e
n
H)+̂(Σ n K))
.
Indeed, this shows that (Σ n H)+̂(Σ n K) is a self-adjoint operator on Λn H0 and Σ n (H +̂K)
= (Σ n H)+̂(Σ n K).
Theorem 4.9. If H ∈ B(H) is self-adjoint and K is a lower-bounded self-adjoint
operator on H, then
(e−rH #α e−rK )1/r ≺w(log) e−((1−α)H+αK) ,
r > 0.
P r o o f. Lemmas 3.1 and 4.8 imply that
Λn ((e−rH #α e−rK )1/r ) = (Λn (e−rH ) #α Λn (e−rK ))1/r = (e−rΣ
Λn (e−((1−α)H+αK) ) = e−((1−α)Σ
n
n
H+αΣ K)
n
H
#α e−rΣ
n
K 1/r
)
,
.
So, as in the proof of Theorem 4.2, we may show that
k(e−rH #α e−rK )1/r k∞ ≤ ke−((1−α)H+αK) k∞ .
Thus it suffices to show that if e−((1−α)H+αK) ≤ I, i.e. (1 − α)H + αK ≥ 0, then
e−rH #α e−rK ≤ I for all r > 0. But this is (i)⇒(ii) of Proposition 4.6.
Problem 4.10. It is desirable to prove Theorem 4.9 when both H and K are lowerbounded. In view of Lemma 4.8, the problem consists in proving that (1 − α)H +̂αK ≥ 0
implies e−tH #α e−tK ≤ I for t ≥ 0.
4.3. Exponential product formula for operator means. The next theorem is a Trotterlike product formula for operator means.
Theorem 4.11. Let σ be an operator mean with α = f ′ (1) for the corresponding
operator monotone function f . If H ∈ B(H) is self-adjoint and K is a lower-bounded
self-adjoint operator on H, then
s-lim(e−rtH σ e−rtK )1/r = e−t((1−α)H+αK) ,
r↓0
t > 0,
the convergence being uniform in t ∈ [a, b] for any 0 < a < b.
We divide the proof of the theorem into several lemmas.
Lemma 4.12. Let G(t), G1 (t), and G2 (t) be SOT-continuous functions from [0, ∞) to
B(H) such that G(t), G1 (t), and G2 (t) mutually commute for any t ≥ 0 and such that
for some a ≥ 0,
0 ≤ G1 (t) ≤ G(t) ≤ G2 (t) ≤ eat I,
t ≥ 0.
Let S be a self-adjoint operator on H and D a core of S. If
Gi (t) − I
lim ξ + Sξ ξ ∈ D, i = 1, 2,
= 0,
t↓0
t
LOG-MAJORIZATIONS AND NORM INEQUALITIES
then
G(t) − I
lim ξ + Sξ = 0,
t↓0
t
165
ξ ∈ D.
P r o o f. Replacing G(t), G1 (t), G2 (t) by those with multiple e−at and S by S + aI, it
suffices to show the case a = 0. Let ξ ∈ D. Since by assumption
I − G(t)
I − G1 (t)
I − G2 (t)
≤
≤
,
0≤
t
t
t
we get
I − G2 (t)
I − G(t)
I − G1 (t)
ξ, ξ ≤
ξ, ξ ≤
ξ, ξ .
t
t
t
This implies that ht−1 (I − G(t))ξ, ξi → hSξ, ξi as t ↓ 0. By polarization we have
I − G(t)
(4.12)
ξ, η → hSξ, ηi,
η ∈ D.
t
Thanks to the commuting assumption of G(t), G1 (t), G2 (t), we also get
2 2 2
I − G(t)
I − G1 (t)
I − G2 (t)
≤
≤
,
t
t
t
so that
I − G2 (t) I − G(t) I − G1 (t) ξ ≤ ξ ≤ ξ
.
t
t
t
Hence kt−1 (I −G(t))ξk → kSξk and t−1 (I −G(t))ξ, t > 0, are bounded. These and (4.12)
imply that t−1 (I − G(t))ξ → Sξ weakly. Therefore
I − G(t)
I − G(t) 2
I − G(t)
ξ − Sξ = ξ − 2Re
ξ, Sξ + kSξk2
t
t
t
showing the conclusion.
→ kSξk2 − 2hSξ, Sξi + kSξk2 = 0,
Let H, K be as in Theorem 4.11 and define
F (t) = e−tH σ e−tK ,
t ≥ 0.
Let L = (1 − α)H + αK and D = D(K), which is the domain of L. Then:
Lemma 4.13.
F (t) − I
ξ + Lξ lim = 0,
t↓0
t
ξ ∈ D.
P r o o f. Define A(t) = etH/2 e−tK etH/2 , so that F (t) = e−tH/2 f (A(t))e−tH/2 for t ≥ 0.
Choose a > 0 such that H ≤ aI and K + aI ≥ 0. We have
F (t) − I
e−tH/2 − I
f (A(t)) − I
e−tH/2 − I
ξ = e−tH/2 f (A(t))
ξ + e−tH/2
ξ+
ξ.
t
t
t
t
Since A(t) → I in SOT and hence f (A(t)) → I in SOT, the first and third terms in the
right-hand side of (4.13) strongly converge as t ↓ 0 to (−H/2)ξ, so that what remains to
show is
f (A(t)) − I
ξ + α(K − H)ξ ξ ∈ D.
(4.14)
lim = 0,
t↓0
t
(4.13)
166
F. HIAI
Define G(t) = f (A(t)), G1 (t) = A(t)((1 − α)A(t) + αI)−1 , and G2 (t) = (1 − α)I + αA(t),
which mutually commute for each t ≥ 0. Since G2 (t) ≤ e2at I and (4.2) yields 0 ≤ G1 (t) ≤
G(t) ≤ G2 (t), we can apply Lemma 4.12. Let ξ ∈ D. It is easy to check that
G2 (t) − I
= α A(t) − I ξ + (K − H)ξ → 0.
ξ
+
α(K
−
H)ξ
t
t
Furthermore, we get
G1 (t) − I
ξ + α(K − H)ξ t
−1 A(t) − ((1 − α)A(t) + αI)
≤ ((1 − α)A(t) + αI)
ξ + α(K − H)ξ t
+ k{((1 − α)A(t) + αI)−1 − I}α(K − H)ξk
A(t) − I
+ (1 − α)k(A(t) − I)(K − H)ξk → 0
ξ
+
(K
−
H)ξ
≤
t
thanks to k((1 − α)A(t) + αI)−1 k∞ ≤ α−1 . Hence Lemma 4.12 implies (4.14).
To prove the theorem, we may assume H, K ≥ 0 by taking H + aI, K + aI, so that
0 ≤ F (t) ≤ I for all t ≥ 0. Now let us proceed as in [19]. Fix any sequence {sn } such
that 0 < sn → ∞. Let Ln = sn (I − F (s−1
n )) so that Ln ≥ 0. Then Lemma 4.13 says that
k(L − Ln )ξk → 0 for all ξ ∈ D.
Lemma 4.14.
sn
− e−Ln ) = 0.
s-lim(F (s−1
n )
n→∞
P r o o f. For any ξ ∈ H we have
sn
kF (s−1
n ) ξ
−e
−Ln
∞
X
skn
−1 k −sn
−1 sn
F (sn ) ξ ξk = F (sn ) ξ − e
k!
k=0
≤ e−sn
≤ e−sn
Since
r
0 ≤ I − F (s−1
n ) ≤
it follows that
∞
X
sk
n
k=0
∞
X
k=0
k!
k
sn
− F (s−1
k(F (s−1
n ) )ξk
n )
skn
|k−sn |
k(I − F (s−1
)ξk.
n )
k!
r(I − F (s−1
n )),
I − F (s−1
n ),
kF (sn−1 )sn ξ − e−Ln ξk ≤ e−sn k(I − F (s−1
n ))ξk
r ≥ 1,
0 ≤ r ≤ 1,
∞
X
skn
|k − sn | + k(I − F (s−1
n ))ξk.
k!
k=0
The Schwarz inequality gives
1/2 X
1/2
X
∞
∞
∞
X
skn
skn
skn
2
sn
|k − sn | ≤
(k − sn )
= esn /2 (sn esn )1/2 = s1/2
n e .
k!
k!
k!
k=0
k=0
k=0
LOG-MAJORIZATIONS AND NORM INEQUALITIES
Hence for every ξ ∈ D,
167
1/2
−1
−Ln
sn
ξk ≤ (s1/2
kF (s−1
n + 1)k(I − F (sn ))ξk =
n ) ξ−e
sn + 1
kLn ξk → 0.
sn
sn
and e−Ln are contractions.
This shows the assertion because F (s−1
n )
The next step is much simpler than that in [19], because the functional calculus can
be applied in our setting.
Lemma 4.15.
s-lim e−Ln = e−L .
n→∞
P r o o f. First note that 0 ≤ (I + Ln )−1 ≤ I and 0 ≤ (I + L)−1 ≤ I. If η = (I + L)ξ
with ξ ∈ D, then
k(I + Ln )−1 η − (I + L)−1 ηk = k(I + Ln )−1 {(I + Ln )ξ + (L − Ln )ξ} − ξk
= k(I + Ln )−1 (L − Ln )ξk ≤ k(L − Ln )ξk → 0.
This implies that (I + Ln )−1 → (I + L)−1 in SOT. Now the result follows due to the
functional calculus as in the proof of Lemma 4.8(1).
P r o o f o f T h e o r e m 4.11 (Completion). Lemmas 4.14 and 4.15 show that
sn
F (s−1
→ e−L in SOT if 0 < sn → ∞; namely
n )
(4.15)
s-lim F (rn )1/rn = e−L if 0 < rn → 0.
n→∞
Replacing H, K by tH, tK with t > 0 yields
s-lim(e−rtH σ e−rtK )1/r = e−tL ,
r↓0
t > 0.
Finally it is an easy task to show the uniform convergence. Indeed, let 0 < a < b and
suppose that the above convergence is not uniform in t ∈ [a, b]. Then there exist ξ ∈ H,
ε > 0, rn ↓ 0, and tn ∈ [a, b] such that
(4.16)
kF (rn tn )1/rn ξ − e−tn L ξk ≥ ε,
n ∈ N.
Choosing a subsequence we may assume that tn → t. Then since λtn → λt uniformly in
λ ∈ [0, 1], it follows from (4.15) that F (rn tn )1/rn = (F (rn tn )1/rn tn )tn → e−tL in SOT as
well as ke−tn L − e−tL k∞ → 0, contradicting (4.16).
It would be expected that Theorem 4.9 is a consequence of Theorems 4.2 and 4.11.
Qn
However this is not truly the case, because i=1 µi (A) is not continuous in SOT.
It is worth noting the following specialization to the matrix case.
Corollary 4.16. For arbitrary Hermitian matrices H and K,
d tH
e σ etK = (1 − α)H + αK,
dt
t=0
lim (etH σ etK )1/t = e(1−α)H+αK ,
t→0
d
log(etH σ etK )
= (1 − α)H + αK.
dt
t=0
P r o o f. Theorem 4.11 gives the second, while the first follows from Lemma 4.13. For
the third, take the logarithm of the second.
168
F. HIAI
R e m a r k 4.17. Theorem 4.11 fails to hold when both H and K are lower-bounded.
Indeed, Kosaki [56] gave an example of positive self-adjoint operators H, K such that
D(H 1/2 ) ∩ D(K 1/2 ) is dense in H and hence e−(H +̂K)/2 is non-zero, but e−rH : e−rK = 0
for each r > 0.
4.4. Norm inequalities of complementary Golden–Thompson type. Theorems 4.9 and
4.11 show:
Corollary 4.18. Let σ be an operator mean such that σ ≤ #α , i.e. f (x) ≤ xα for
x ≥ 0 where f is the corresponding operator monotone function. If H and K are as in
Theorem 4.11 and k · k is a symmetric norm, then
k(e−rH σ e−rK )1/r k ≤ ke−((1−α)H+αK) k,
(4.17)
r > 0,
and moreover
(4.18)
lim k(e−rH σ e−rK )1/r k = ke−((1−α)H+αK) k.
r↓0
If σ = #α then the left-hand side of (4.17) increases to ke−((1−α)+αK) k as r ↓ 0.
P r o o f. Since e−rH σ e−rK ≤ e−rH #α e−rK by assumption, Theorem 4.9 gives for
every r > 0
k(e−rH σ e−rK )1/r k ≤ k(e−rH #α e−rK )1/r k ≤ ke−((1−α)H+αK) k.
On the other hand, in view of Proposition 2.11, Theorem 4.11 implies that
ke−((1−α)H+αK) k ≤ lim inf k(e−rH σ e−rK )1/r k.
r↓0
Therefore the first part is shown. The second part is immediate from (4.4).
In particular, if σ is a symmetric operator mean with σ ≤ # and H, K are as above,
then
(4.19)
k(e−2rH σ e−2rK )1/r k ≤ ke−(H+K) k,
r > 0,
−(H+K)
and the left-hand side converges to ke
k as r ↓ 0. For instance, when σ = ! or
σ = #, the left-hand side of (4.19) increases to the right as r ↓ 0. Norm inequalities such
as (4.17) and (4.19) are considered as complementary to Golden–Thompson ones (see [6,
44, 71]).
R e m a r k s 4.19. (1) Let H = 0, K = xI for x ∈ R, and k · k = k · k∞ . Then (4.17)
menas that f (e−rx ) ≤ e−αrx for r > 0 and so σ ≤ #α . Thus the assumption σ ≤ #α is
essential in Corollary 4.18.
(2) Inequality (4.19) does not hold for a general symmetric mean. Indeed, if e−K ∈
Cp (H) where 0 < p < ∞ then, for any bounded H, e−(H+K) ∈ Cp (H) by (3.11), but
k(e−2rH ∇ e−2rK )1/r kp = ∞ for all r > 0.
When we are concerned with (quasi-)norms k · kp , the following holds:
Proposition 4.20. Let σ be as in Corollary 4.18. Let H and K be lower-bounded
self-adjoint operators such that H + K is essentially bounded. If e−K ∈ Cp (H) where
0 < p < ∞, then
k(e−rH/(1−α) σ e−rK/α )1/r kp ≤ ke−(H+K) kp ,
r > 0.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
169
P r o o f. The case p = 1 is enough because we may replace H, K by pH, pK (see
−K
−(H+K)
the proof of Corollary
∈ C1 (H) by
Tn 3.10). So assume that e ∈ C1 (H). Hence e
(3.11). Let Hn = a λdEH (λ). Then we proved the convergence (3.22). For each r > 0,
since e−rH/(1−α) ≤ e−rHn /(1−α) , we get
k(e−rH/(1−α) σ e−rK/α )1/r k1 ≤ k(e−rHn /(1−α) σ e−rK/α )1/r k1 ≤ ke−(Hn +K) k1
by (4.17). This and (3.22) yield the conclusion.
4.5. Norm convergence of exponential product formula for operator means. In this
subsection let us consider norm convergence of (e−rH/(1−α) σ e−rK/α )1/r as r ↓ 0 when
k · k is uniformly convex or k · k = k · kp for 0 < p < ∞.
In Banach space theory, there are several important geometric notions of convexity
or smoothness for norms (see e.g. [13]). For instance, a Banach space X is said to be
uniformly convex if for each ε > 0 there exists δ > 0 such that, for any x, y ∈ X ,
k(x + y)/2k ≤ 1 − δ holds whenever kxk = kyk = 1 and kx − yk ≥ ε. A Hilbert space
is a typical example of a uniformly convex Banach space. As is well known, a uniformly
convex Banach space is reflexive. A useful property of a uniformly convex space X is
that if {xj } ⊂ X weakly converges to x ∈ X and kxj k → kxk, then kxj − xk → 0. This
property is typical in a Hilbert space, as was already used in the proof of Theorem 3.12.
When 1 < p < ∞, the uniform convexity of Cp (H) is an immediate consequence of
the Clarkson–McCarthy inequalities:
kA + Bkpp + kA − Bkpp ≤ 2p−1 (kAkpp + kBkpp ),
2 ≤ p < ∞,
kA + Bkqp + kA − Bkqp ≤ 2(kAkpp + kBkpp )q/p ,
1 < p ≤ 2,
where 1/p + 1/q = 1. (The proof in the most general setup is found in [26].)
Now let Φ be a symmetric gauge function, Φ′ the conjugate one, and k·k the symmetric
norm corresponding to Φ. Assume that CΦ (H) (or k · k) is uniformly convex. Then the
reflexivity of CΦ (H) implies as remarked after Theorem 2.9 that both Φ and Φ′ are regular
and so CΦ (H)∗ ∼
= CΦ′ (H).
Lemma 4.21. Assume that CΦ (H) is uniformly convex. If {Aj } ⊂ CΦ (H), supj kAj k <
∞, and Aj → A in WOT , then Aj → A in w(CΦ (H), CΦ′ (H)), the weak topology.
P r o o f. First the assumptions give A ∈ CΦ (H) by Proposition 2.11. Since Φ′ is regular
as remarked above, Cfin (H) is dense in CΦ′ (H). By the WOT-convergence of {Aj } we get
tr((Aj − A)B) → 0 for any B ∈ Cfin (H). In view of Theorem 2.9, this and the k · kboundedness imply the conclusion.
Corollary 4.22. Let σ be as in Corollary 4.18. Assume that k·k is uniformly convex.
If H and K are as in Theorem 4.11 and e−K ∈ CΦ (H), then
lim k(e−rH/(1−α) σ e−rK/α )1/r − e−(H+K) k = 0.
r↓0
P r o o f. By (3.11), (4.17), Theorem 4.11, and Lemma 4.21 altogether, we have as
r ↓ 0,
(e−rH/(1−α) σ e−rK/α )1/r → e−(H+K) in w(CΦ (H), CΦ′ (H)).
Hence the result follows from the uniform convexity and (4.18).
170
F. HIAI
Concerning k · kp , 0 < p < ∞, we have:
Corollary 4.23. Let σ be as in Corollary 4.18. If H and K are as in Theorem 4.11
and e−K ∈ Cp (H) where 0 < p < ∞, then
lim k(e−rH/(1−α) σ e−rK/α )1/r − e−(H+K) kp = 0.
r↓0
P r o o f. The case 1 < p < ∞ is included in Corollary 4.22. Let 0 < p ≤ 1 and choose
k ∈ N such that 2k > 1/p. Applying Corollary 4.22 to k · k2k p and 2−k H, 2−k K, we have
k
lim k(e−rH/(1−α) σ e−rK/α )1/2
(4.20)
k
r
− e−(H+K)/2 k2k p = 0.
r↓0
Noting that kX +Y kq ≤ 21/q (kXkq +kY kq ) for any q > 0, we get by the Hölder inequality
k−1
k(e−rH/(1−α) σ e−rK/α )1/2
k−1
≤ 21/2
p
r
k−1
− e−(H+K)/2
k
{k(e−rH/(1−α) σ e−rK/α )1/2
k
× [(e−rH/(1−α) σ e−rK/α )1/2
k
k−1
p
r
k
r
+ k[(e−rH/(1−α) σ e−rK/α )1/2
≤ 21/2
k2k−1 p
− e−(H+K)/2 ]k2k−1 p
r
k
k
− e−(H+K)/2 ]e−(H+K)/2 k2k−1 p }
k
k(e−rH/(1−α) σ e−rK/α )1/2
r
k
− e−(H+K)/2 k2k p
k
k
× {k(e−rH/(1−α) σ e−rK/α )1/2 r k2k p + ke−(H+K)/2 k2k p }.
This implies by (4.20) and Proposition 4.20 that
k−1
lim k(e−rH/(1−α) σ e−rK/α )1/2
r↓0
r
k−1
− e−(H+K)/2
k2k−1 p = 0.
Repeating this argument yields the result.
5. Miscellaneous results. In this section we first discuss the interplay between
log-majorizations and the Furuta inequalities [28, 30]. Several log-majorization results
are obtained in Subsection 5.2. Furthermore, we give determinant inequalities as simple
applications of the log-majorization results.
5.1. Interplay between log-majorization and Furuta inequlity. The Furuta inequality
[28] says that if A ≥ B ≥ 0 in B(H)+ , then
(B r/2 Ap B r/2 )α ≥ B α(p+r) ,
Aα(p+r) ≥ (Ar/2 B p Ar/2 )α ,
(5.1)
whenever 0 ≤ α ≤ 1, p, r ≥ 0, and 1 + r ≥ α(p + r).
As was clarified in [6], we can transform Furuta type operator inequalities into logmajorizations and vice versa. For instance, the Furuta inequality (5.1) is reformulated in
terms of log-majorization as follows.
Proposition 5.1. For every A, B ∈ B(H)+ , if 0 < α < 1, p ≥ 0, and 0 ≤ r ≤
min{α, αp}, then
p−r
µ(A 1−α #α (Ar/2α B p/α Ar/2α )) ≺w(log) µ((A1/2 BA1/2 )p ).
Furthermore, the assumption r ≥ 0 is removed when A is invertible.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
171
P r o o f. We may assume that A is invertible. Let 0 < α < 1, q, s ≥ 0, and 1 + s ≥
α(q + s). Then it is seen from (5.1) that A1/2 BA1/2 ≤ I or A−1 ≥ B implies
Aα(q+s) #α (A(α(q+s)−s)/2 B q A(α(q+s)−s)/2 ) = Aα(q+s)/2 (A−s/2 B q A−s/2 )α Aα(q+s)/2 ≤ I.
Arranging the order of homogeneity, we have
kAα(q+s) #α (A(α(q+s)−s)/2 B q A(α(q+s)−s)/2 )k∞ ≤ k(A1/2 BA1/2 )αq k∞ .
As in the proofs of Theorems 3.4 and 4.2, the antisymmetric tensor technique yields
µ(Aα(q+s) #α (A(α(q+s)−s)/2 B q A(α(q+s)−s)/2 )) ≺w(log) µ((A1/2 BA1/2 )αq ).
Now putting p = αq and r = α(α(q+s)−s), we get the conclusion, where the assumptions
q, s ≥ 0 and 1 + s ≥ α(q + s) are transformed into p ≥ 0 and r ≤ min{α, αp}.
Corollary 5.2. If A, B ∈ B(H)+ and A is invertible, then for every 0 ≤ α ≤ 1,
µ(Aα (A−1/2 BA−1/2 )α Aα ) ≺w(log) µ(Aα/2 B α Aα/2 ) ≺w(log) µ((A1/2 BA1/2 )α ).
P r o o f. The case 0 < α < 1 is enough. The second log-majorization is nothing but
(3.6). For the first, take p = 1 and r = 2α − 1 in Proposition 5.1. Then we have
µ(A(A−1/2α B 1/α A−1/2α )α A) ≺w(log) µ(A1/2 BA1/2 ),
which gives the result by replacing A, B by Aα , B α , respectively.
Also, we can consider the converse direction from log-majorizations to operator inequalities. For instance, the following is a reformulation of the log-majorization (4.3).
Proposition 5.3. If A ≥ B ≥ 0 and A is invertible, then
(5.2)
Ar ≥ {Ar/2 (A−1/2 B p A−1/2 )r Ar/2 }1/p ,
p, r ≥ 1.
P r o o f. Let A, B ∈ B(H)+ with A invertible. Let 0 < α ≤ 1. Then (4.3) says that
A ≥ (A−1/2 BA−1/2 )α implies A−r ≥ (A−r/2 B r A−r/2 )α for every r ≥ 1. Put p = 1/α
and replace A−1 , (A−1/2 BA−1/2 )α by A, B, respectively. Then the result follows.
−1
Recently Furuta [30] obtained a quite general operator inequality extending the original Furuta inequality as well as (5.2), which is stated in the following:
Theorem 5.4. Assume that A ≥ B ≥ 0 and A is invertible. Then
(5.3)
1−t+r
A1−t+r ≥ {Ar/2 (A−t/2 B p A−t/2 )s Ar/2 } (p−t)s+r ,
whenever 0 ≤ t ≤ 1, p ≥ 1, rget, and s ≥ 1. Furthermore, for every 0 ≤ t ≤ 1 and p ≥ 1,
1−t+r
(r, s) 7→ A−r/2 {Ar/2 (A−t/2 B p A−t/2 )s Ar/2 } (p−t)s+r A−r/2
is a decreasing function of both r ≥ t and s ≥ 1.
Note that (5.1) is a consequence of (5.3) in case of t = 0 and s = 1. Also (5.2) is a
special case of (5.3) when t = 1 and r = s ≥ 1. In this way, the operator inequality (5.3)
interpolates (5.1) and (5.2) by the parameter 0 ≤ t ≤ 1.
In [30] several log-majorizations extending (4.3) were produced from (5.3) in the same
way as in the proof of Proposition 5.1. Among others, the following is obtained from (5.3)
by taking t = 1 and p = 1/α. But for A compact, see the proof of Theorem 4.2.
172
F. HIAI
Theorem 5.5. If A, B ∈ B(H)+ and A is either invertible or compact , then for every
0 ≤ α ≤ 1 and r, s ≥ 1
µ(Ar #αq/s B s ) ≺w(log) µ((A #α B)q )
where q = ((1 − α)r−1 + αs−1 )−1 .
5.2. Other log-majorizations. In this subsection let us show some other log-majorization results. The first two propositions were given in [6] in the matrix case, while the
remaining are new.
For any T ∈ B(H), the real part of T is Re T = (T + T ∗ )/2, and the exponential
P∞
operator eT is defined as the power series expansion eT = n=0 T n /n!. The following
log-majorization (1) for exponential operator is essentially due to [14].
Proposition 5.6. Let T ∈ B(H).
(1) µ(eT ) ≺w(log) µ(eRe T ).
(2) Assume that Re T ≥ 0. Then |eT | ≥ I (or log |eT | ≥ 0) and µ(log |eT |) ≺w
µ(Re T ). Hence k log |eT | k ≤ kRe T k for any symmetric norm k · k.
P r o o f. (1) For every X ∈ B(H) we get by (1.14)
µ(X k ) ≺w(log) µ(X)k ,
Putting X = eT /k in the above yields
∗
k ∈ N.
µ(eT ) ≺w(log) µ(|eT /k |k ),
k ∈ N.
But |eT /2k |2k = (eT /2k eT /2k )k converges in the norm k · k∞ to e(T
k → ∞. Hence the assertion is shown by Proposition 1.4(10).
(2) If Re T ≥ 0, then since
∗
∗
∗
+T )/2
= eRe T as
∗
|eT |−1 = (eT eT )−1/2 = (e−T e−T )1/2 = |e−T |,
we get by (1),
∗
k |eT |−1 k∞ ≤ keRe(−T ) k∞ = ke−Re T k∞ ≤ 1,
so that |eT | ≥ I. Then the required weak majorization is immediate from (1), because
µn (log |eT |) = log µn (eT ) and µn (Re T ) = log µn (eRe T ).
Proposition 5.7. Let A, B ∈ B(H)+ .
(1) µ(Ap1 B q1 Ap2 B q2 . . . Apk B qk ) ≺w(log) µ(Ap1 +...+pk B q1 +...+qk ) if 0 ≤ p1 ≤ q1 ≤
p1 + p2 ≤ q1 + q2 ≤ . . . ≤ q1 + . . . + qk−1 ≤ p1 + . . . + pk = q1 + . . . + qk (in particular , if
pj = qj ≥ 0 for all j).
(2) µ(Ap B q Ar ) ≺w(log) µ(Ap+r B q ) for every p, q, r ≥ 0.
(3) µ((AB)k A) ≺w(log) µ(A(k+1)/2 B k A(k+1)/2 ) for every k ∈ N.
P r o o f. (1) We may assume that p1 + . . . + pk = q1 + . . . + qk = 1. It suffices as in
the proof of Theorem 3.4 to show that when A, B are invertible, kABk∞ ≤ 1 implies
kAp1 B q1 . . . Apk B qk k∞ ≤ 1. Suppose kABk∞ ≤ 1 or BA2 B ≤ I. Then A2 ≤ B −2 and
B 2 ≤ A−2 . Since 0 ≤ q1 − p1 ≤ 1, we get
B q1 A2p1 B q1 ≤ B q1 B −2p1 B q1 = B 2(q1 −p1 ) ≤ A2(p1 −q1 )
173
LOG-MAJORIZATIONS AND NORM INEQUALITIES
and hence
Ap2 B q1 A2p1 B q1 Ap2 ≤ A2(p1 +p2 −q1 ) ≤ B 2(q1 −p1 −p2 ) .
Repeating this argument yields
B qk Apk . . . B q1 A2p1 B q1 . . . Apk B qk ≤ B 2(q1 +...+qk −p1 −...−pk ) = I,
so that kAp1 B q1 . . . Apk B qk k∞ ≤ 1.
(2) Put p1 = p, p2 = r, q1 = p + r, and q2 = 0 in (1). Then
q
q
µ(Ap B q Ar ) = µ(Ap (B p+r )p+r Ar ) ≺w(log) µ(Ap+r (B p+r )p+r ) = µ(Ap+r B q ).
(3) When k = 2m − 1 we get
µ((AB)2m−1 A) = µ((AB)m−1 AB 1/2 )2
2m
= µ({(A 2m−1 )
2m−1
2m
2m
B}m−1 (A 2m−1 )
2m−1
2m
B 1/2 )2
≺w(log) µ(Am B (2m−1)/2 )2 = µ(Am B 2m−1 Am ).
The log-majorization in the above follows from (1) with
2m − 1
p1 = . . . = pm =
, q1 = . . . = qm−1 = 1,
2m
The derivation is similar when k = 2m.
qm =
1
.
2
Let H and K be lower-bounded self-adjoint operators on H. For t > 0 put S(t) =
e−tH/2 e−tK e−tH/2 , which is the second order approximant of e−t(H +̂K) .
Then
{S(t/n)S((1 − 2t)/n)S(t/n)}n and {S(t/n)2 S((1 − 4t)/n)S(t/n)2 }n are sometimes used
P
as higher order approximants of e−(H +̂K) . When p1 , . . . , pk ≥ 0 and kj=1 pj = 1, Proposition 5.7(1) gives
µ(S(p1 )S(p2 ) . . . S(pk )) ≺w(log) µ(e−H e−K ).
A special case of the norm inequality proved by Kato [50] is the following:
r
kAr XB r k∞ ≤ kXk1−r
∞ kAXBk∞ ,
(5.4)
0 ≤ r ≤ 1,
for every A, B ∈ B(H)+ and X ∈ B(H). For the proof of (5.4), it is enough to show that
log kAr XB r k∞ is convex in r ∈ [0, 1]. This follows because for 0 ≤ r ≤ s ≤ 1,
kA
r+s
2
XB
r+s
2
k2∞ = r(B
r+s
2
X ∗ Ar+s XB
r+s
2
) = r(B r X ∗ Ar As XB s )
≤ kAr XB r k∞ kAs XB s k∞ .
Applying the antisymmetric tensor technique to (5.4), we have the following generalization of Theorem 3.4.
Proposition 5.8. For every A, B ∈ B(H)+ and X ∈ B(H),
(5.5)
µ(Ar XB r ) ≺w(log) µ(X)1−r µ(AXB)r ,
0 ≤ r ≤ 1,
or equivalently
µ(AXB)r ≺w(log) µ(X)r−1 µ(Ar XB r ),
r ≥ 1.
This yields the norm inequality proved by Kittaneh [52] as follows.
Corollary 5.9. Let A, B ∈ B(H)+ , X ∈ B(H), and k · k be a symmetric norm. Then
(5.6)
kAr XB r k ≤ kXk1−r kAXBkr ,
0 ≤ r ≤ 1.
174
F. HIAI
P r o o f. Let Φ be the corresponding gauge function. For each 0 < r < 1, we have by
Proposition 1.3 and by Lemmas 2.2 and 2.13(1)
kAr XB r k ≤ Φ(µ(X)1−r µ(AXB)r ) ≤ Φ(µ(X))1−r Φ(µ(AXB))r = kXk1−r kAXBkr .
We have also
µ(Ar XB 1−r ) ≺w(log) µ(AX)r µ(XB)1−r ,
(5.7)
0 ≤ r ≤ 1,
modifying (5.5) in the same way as in [52] where it is observed that (5.6) is equivalently
reformulated as
kAr XB 1−r k ≤ kAXkr kXBk1−r ,
0 ≤ r ≤ 1.
In fact, to check (5.7), we may assume that B is invertible. Then for 0 ≤ r ≤ 1 we get
by (5.5)
µ(Ar XB 1−r ) = µ(Ar (XB)B −r ) ≺w(log) µ(XB)1−r µ(A(XB)B −1 )r
= µ(AX)r µ(XB)1−r .
Furthermore, for any A, B, X ∈ B(H), it is immediate from polar decompositions that
µ(AXB) = µ(|A| · X · |B ∗ |). Hence (5.7) with r = 1/2 yields
µ(AXB) ≺w(log) µ(A∗ AX)1/2 µ(XBB ∗ )1/2 ,
(5.8)
which is a generalization of Horn’s majorization (1.14). Let r > 0, 1 < p, q < ∞ with
1/p + 1/q = 1, and k · k be a symmetric norm. Then by (5.8) and Lemma 2.13(1) we have
k |AXB|r k ≤ k |A∗ AX|rp/2 k1/p k |XBB ∗ |rq/2 k1/q .
In particular,
k |AB|r k ≤ k |A|rp k1/p k |B|rq k1/q ,
k |AXB|r k2 ≤ k |A∗ AX|r k · k |XBB ∗ |r k.
The last inequality was given in [16].
As is well known, the Golden–Thompson trace inequality fails to hold for three matrices, while an attempt in the case of three matrices was made in [59]. In the rest of this
subsection, we discuss log-majorizations of Golden–Thompson type for more than two
operators which are commuting except one.
Proposition 5.10. (1) If A1 , A2 ∈ B(H)+ and A1 A2 = A2 A1 , then for every B ∈
B(H),
µ(BA1 A2 B ∗ ) = µ((A1 A2 )1/2 B ∗ B(A1 A2 )1/2 ) ≺w(log) µ(A1 B ∗ BA2 ).
(2) If A1 , A2 , A3 ∈ B(H)+ and Ai Aj = Aj Ai , then for every B ∈ B(H),
µ(BA1 A2 A3 B ∗ ) ≺w(log) µ(A21 B ∗ BA22 B ∗ BA23 )1/2 .
P r o o f. (1) The equality is obvious because BA1 A2 B ∗ = |(A1 A2 )1/2 B ∗ |2 and
(A1 A2 )1/2 B ∗ B(A1 A2 )1/2 = |B(A1 A2 )1/2 |2 .
For the log-majorization, by the antisymmetric tensor technique, it suffices to show
that |A1 B ∗ BA2 | ≤ I implies (A1 A2 )1/2 B ∗ B(A1 A2 )1/2 ≤ I. We may assume that A2 is in∗
2 ∗
vertible. Then A2 B ∗ BA21 B ∗ BA2 ≤ I implies B ∗ BA21 B ∗ B ≤ A−2
2 and so A1 B BA1 B BA1
−1 2
−1
∗
1/2 ∗
1/2
≤ (A1 A2 ) . This gives A1 B BA1 ≤ A1 A2 and hence (A1 A2 ) B B(A1 A2 )
≤ I.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
175
(2) By using (1) twice, we have
µ(BA1 A2 A3 B ∗ ) ≺w(log) µ(A2 B ∗ BA1 A3 ) = µ(A1 A3 (A2 B ∗ B)∗ (A2 B ∗ B)A1 A3 )1/2
≺w(log) µ(A21 B ∗ BA22 B ∗ BA23 )1/2 .
Proposition 5.11. Let H1 , H2 , H3 , K be lower-bounded self-adjoint operators on H.
Assume that H1 , H2 , H3 are mutually commuting (i.e. their spectral projections are all
commuting). Then
(5.9)
(5.10)
µ(e−((H1 +H2 )+̂K) ) ≺w(log) µ(e−H1 e−K e−H2 ),
µ(e−((H1 +H2 +H3 )+̂2K) ) ≺w(log) µ(e−H1 e−K e−H2 e−K e−H3 ).
Hence ke−(H1 +H2 +H3 )+̂2K) k ≤ ke−H1 e−K e−H2 e−K e−H3 k for any symmetric norm k · k.
P r o o f. (5.9) is a special case of (5.10). The commuting assumption guarantees that
H1 + H2 + H3 is defined as a lower-bounded self-adjoint operator and
e−r(H1 +H2 +H3 ) = e−rH1 e−rH2 e−rH3 ,
r ≥ 0.
By (3.6) and Proposition 5.10(2) we have for every 0 < r ≤ 1/2,
µ((e−rK e−r(H1 +H2 +H3 ) e−rK )1/r ) ≺w(log) µ(e−K/2 e−(H1 +H2 +H3 )/2 e−K/2 )2
≺w(log) µ(e−H1 e−K e−H2 e−K e−H3 ).
Hence (5.10) follows by letting r ↓ 0 as in the proof of Theorem 3.7.
Let A1 , . . . , A4 ∈ B(H)+ with Ai Aj = Aj Ai , and B ∈ B(H). Then, using Proposition
5.10(2) twice, we have
µ(BA1 A2 A3 A4 B ∗ ) ≺w(log) µ(A4 B ∗ BA21 A22 A23 B ∗ BA4 )1/2
≺w(log) µ(A41 B ∗ BA24 B ∗ BA42 B ∗ BA24 B ∗ BA43 )1/4 .
This implies as Proposition 5.11 that if H1 , . . . , H4 , K are lower-bounded self-adjoint
operators and H1 , . . . , H4 are mutually commuting, then
µ(e−(H1 +...+H4 )+̂4K) ) ≺w(log) µ(e−H1 e−K e−H4 /2 e−K e−H2 e−K e−H4 /2 e−K e−H3 ).
Problem 5.12. Do (5.9) and (5.10) extend to a greater number of operators? That
is, if H1 , . . . , Hn , K are lower-bounded self-adjoint operators with H1 , . . . , Hn mutually
commuting, then
µ(e−((H1 +...+Hn )+̂(n−1)K) ) ≺w(log) µ(e−H1 e−K e−H2 . . . e−K e−Hn )?
5.3. Determinant inequalities. In the final subsection, some determinant inequalities
of Golden–Thompson type are given. Let A ∈ C1 (H). Following [32], we define the determinant det(I + A) by
∞
Y
det(I + A) =
(1 + λk (A)),
k=1
where λk (A) are the eigenvalues of A counted with algebraic multiplicities. The product
of the above right-hand side converges because Theorem 3.3 and Proposition 1.3 imply
that
∞
∞
X
X
|λk (A)| ≤
µk (A) = kAk1 < ∞.
k=1
k=1
176
F. HIAI
Another equivalent definition is
(5.11)
det(I + A) =
∞
X
tr(Λn A).
n=0
Indeed, note that if A ∈ C1 (H) then kΛn Ak1 ≤ kAkn1 /n! for n ∈ N and
det(I + zA) =
∞
X
z n tr(Λn A),
n=0
z ∈ C,
is an entire analytic function. In particular, comparing the coefficients of z, we have
Lidskii’s theorem:
∞
X
tr A =
λk (A).
k=1
It is known that for every A, B ∈ C1 (H),
(5.12)
|det(I + A) − det(I + B)| ≤ kA − Bk1 exp(kAk1 + kBk1 + 1),
which shows that A 7→ det(I +A) is k·k1 -continuous on C1 (H). See [79, §3], [74, §XIII.17],
or [31, VII] for details of the facts on the determinant mentioned above.
Lemma 5.13. Let a = (a1 , a2 , . . .) and b = (b1 , b2 , . . .) be such that a1 ≥ a2 ≥ . . . ≥ 0
Q∞
Q∞
and b1 ≥ b2 ≥ . . . ≥ 0. If a ≺w(log) b, then k=1 (1 + ak ) ≤ k=1 (1 + bk ).
P r o o f. Since log(1 + x) is increasing on [0, ∞) and log(1 + ex ) is convex, Proposition
Qk
Qk
1.3 implies that (log(1 + ai )) ≺w (log(1 + bi )) and hence i=1 (1 + ai ) ≤ i=1 (1 + bi ) for
every k ∈ N. This gives the result.
Proposition 5.14. Let H and K be lower-bounded self-adjoint operators on H. Assume that e−K ∈ C1 (H). Then
(5.13)
det(I + e−(H +̂K) ) ≤ det(I + (e−rH/2 e−rK e−rH/2 )1/r ),
r > 0,
and the above right-hand side decreases as r ↓ 0. In particular ,
(5.14)
det(I + e−(H +̂K) ) ≤ det(I + e−H/2 e−K e−H/2 ) = det(I + e−H e−K ).
Moreover , if H +K is essentially self-adjoint , then the right-hand side of (5.13 ) decreases
to det(I + e−(H+K) ) as r ↓ 0.
P r o o f. Note as remarked after Theorem 3.12 that (e−rH/2 e−rK e−rH/2 )1/r ∈ C1 (H)
for all r > 0. The first assertion is immediate from Theorems 3.4 and 3.7 due to Lemma
5.13. The equality in (5.14) is seen because e−H/2 e−K e−H/2 and e−H e−K have the same
eigenvalues including algebraic multiplicities. (This is seen also via (5.11) and Lemma
3.1.) The last assertion is a consequence of (3.12) and (5.12).
Proposition 5.15. Let σ be an operator mean such that σ ≤ #α where 0 ≤ α ≤ 1.
Let H and K be lower-bounded self-adjoint operators. Assume that H + K is essentially
self-adjoint and e−K ∈ C1 (H). Then
det(I + (e−rH/(1−α) σ e−rK/α )1/r ) ≤ det(I + e−(H+K) ),
r > 0.
Moreover , if H is bounded , then the above left-hand side converges to det(I + e−(H+K) )
as r ↓ 0.
LOG-MAJORIZATIONS AND NORM INEQUALITIES
P r o o f. Let H =
T∞
a
λdEH (λ) and Hn =
Tn
a
177
λdEH (λ). For each r > 0 we get
det(I + (e−rH/(1−α) σ e−rK/α )1/r ) ≤ det(I + (e−rHn /(1−α) #α e−rK/α )1/r )
≤ det(I + e−(Hn +K) )
by Theorem 4.9 and Lemma 5.13. So the desired inequality follows because det(I +
e−(Hn +K) ) → det(I + e−(H+K) ) due to (3.22) and (5.12). If H is bounded, then Theorem
4.11 and Lemma 3.8 show that
n
n
Y
Y
µk (I + (e−rH/(1−α) σ e−rK/α )1/r )
µk (I + e−(H+K) ) ≤ lim inf
r↓0
k=1
k=1
≤ lim inf det(I + (e−rH/(1−α) σ e−rK/α )1/r ).
r↓0
Letting n → ∞ we have the second assertion.
The final result is restricted to the matrix case. Further determinant inequalities for
matrices are found in [6].
Proposition 5.16. Let T be any n × n matrix.
(1) det(I + |eT |) ≤ det(I + eRe T ).
(2) det(log |eT |) ≥ det(Re T ) if Re T ≥ 0.
P r o o f. (1) follows from Proposition 5.6(1) and Lemma 5.13.
(2) For any matrix T , since
det |eT | = |etr T | = etr(Re T ) = det eRe T ,
Proposition 5.6(1) is improved to µ(eT ) ≺(log) µ(eRe T ). Since |eT | ≥ I by Proposition
5.6(2), this means that µ(log |eT |) ≺ µ(Re T ). For any ε > 0, applying Proposition 1.1(iii)
to f (x) = − log(x + ε) we have
n
n
X
X
log{µi (Re T ) + ε},
log{µi (log |eT |) + ε} ≤ −
−
i=1
i=1
so that
n
Y
i=1
{µi (log |eT |) + ε} ≥
n
Y
i=1
{µi (Re T ) + ε}.
Letting ε ↓ 0 yields the required inequality.
Problem 5.17. Is it possible to properly extend the determinant inequalities of Proposition 5.16 to operators? Concerning (1), both sides are equal to∞ for infinite-dimensional
T ∈ B(H), so that we have to consider unbounded T .
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